1  Q 


LIBRARY 

OF  THK 

University  of  California. 

GIFT    OF 

^AJLA: ^M^.Crf 

Class 

ELEMENTS  OF  BUSINESS   ARITHMETIC 


THE  MACMILLAN  COMPANY 

NEW  YORK    •    BOSTON   •    CHICAGO 
ATLANTA   •    SAN   FRANCISCO 

MACMILLAN  &   CO.,  Limited 

LONDON  •  BOMBAY  •  CALCUTTA 
MELBOURNE 

THE  MACMILLAN  CO.  OF  CANADA,  Ltd. 

TORONTO 


MACMILLAN'S    COMMERCIAL    SERIES 

ELEMENTS  OF 

BUSINESS  AEITHMETIC 


BY 

ANSON   H.    BIGELOW 

SUPEBINTBNDENT    CITY   SCHOOLS,    LEAD,    8.1). 
AND 

W.    A.    ARNOLD 

DIRECTOR   BUSINESS    TRAINING,    WOODBINE,    IOWA 
NORMAL   SCHOOL 


THE   MACMILLAN   COMPANY 
1911 

All  rights  reserved 


MAY   5   1911 
GIFT, 


Copyright,  1911, 
By  the  MACMILLAN  COMPANY. 


Set  up  and  electrotyped.     Published  January,  191 1 


NarfaootJ  ^regs 

J.  8.  Cashing  Co.  —  Berwick  &  Smith  Co. 

Norwood,  Mass.,  U.S.A. 


PREFACE 

The  preparation  of  the  text  which  follows  was  undertaken 
in  the  belief  that  the  arithmetic  of  the  schools  should  teach 
the  methods  most  in  vogue  in  the  business  world,  and  that 
those  methods  should  be  so  taught  as  to  form  correct  habits 
in  those  who  are  to  attack  the  problems  of  real  life. 

The  accomplishment  of  these  ends  has  involved,  first,  an 
investigation  into  the  methods  of  the  various  fields  of  business 
activity,  and,  second,  the  writing  of  the  whole  subject  from 
the  point  of  view  of  habit-forming  rather  than  from  that  of 
either  the  conventional  or  the  scientific  treatment. 

The  methods  chosen  are  believed  to  have  the  sanction  of 
usage  by  those  in  the  business  world  best  qualified  to  speak. 
The  manner  of  presentation  has  been  tested  by  nearly  ten 
years  of  use,  in  manuscript  form,  in  the  schools  and  classes 
of  which  the  authors  have  had  charge. 

The  processes  presented  are  strictly  arithmetical.  No 
form  of  domination  by  higher  studies  is  more  insidious  or 
harmful  than  the  attempt  to  apply  the  abstractions  of  algebra 
and  geometry  to  the  problems  of  arithmetic  for  the  use 
of  children.  Immature  young  people  do  not  comprehend 
these  abstractions  and  can  only  memorize  them  and  apply 
them  haltingly.  On  the  other  hand,  if  they  fully  understand 
the  concrete  methods  of  arithmetic  and  can  understandingly 
solve  its  problems,  their  minds  are  better  equipped  with 
those  concrete  concepts  which  alone  give  meaning  to  the 
more  abstract  forms  and  truths  of  the  higher  branches  of 
mathematics. 

While  the  methods  used  in  this  book  are  those  of  the 
counting  room,  the  shop,  or  the  farm,  the  pure  mathematical 

213230 


Vi  PREFACE 

element  has  not  been  sacrificed.  When  the  mathematical 
reasoning  of  available  methods  is  clear,  the  chief  considera- 
tions have  been,  just  as  they  are  in  the  business  world,  short- 
ness of  operation,  quickness  of  solution,  and  the  minimum 
of  opportunity  or  likelihood  of  error.  These  considerations 
have  not  always  led  to  formal  methods,  equally  applicable 
to  all  contingencies,  but  have  rather  pointed  to  direct  ways 
of  solving  the  kind  of  problems  most  frequently  met.  While 
the  emphasis  has  not  been,  therefore,  upon  the  best  methods 
for  infrequent  and  unusual  problems,  their  solution  is  none 
the  less  clearly  prepared  for. 

Topics  admittedly  obsolete  have  been  omitted,  while  others 
less  used  than  formerly  have  received  correspondingly  less 
emphasiSo  In  general,  only  those  topics  or  phases  of  topics 
have  been  treated  which  are  applicable  to  present-day  prob- 
lems, and  in  the  order  of  their  need,  regardless  of  the  tradi- 
tional arrangement  and  sequence  of  subject  matter. 

To  those  who  believe  that  there  should  be  a  re-canvassing 
of  the  whole  field  of  arithmetic  at  the  close  of  the  grammar 
school,  or  in  the  first  high  school  year,  with  more  attention 
to  the  methods  used  in  actual  life  and  with  the  deliberate 
purpose  of  developing  an  habitual  mode  of  attack  which  seeks 
the  most  direct  and  accurate  methods  of  solution,  we  com- 
mend our  book  in  the  hope  that  it  may  be  of  real  service. 
We  are  firmly  convinced  that  there  is  a  large  and  growing 
number  of  school  men  and  women  who  believe  that  this  is 
the  only  road  to  adequate  and  practical  results  from  arith- 
metic teaching. 

We  desire  to  express  appreciation  of  the  sympathetic  criti- 
cism and  cooperation  of  the  editor  of  the  series  in  which 
the  book  appears,  and  of  the  many  courtesies  shown  us  by 
officers  and  employees  of  numerous  large  business  houses 
whom  we  consulted  in  our  quest  for  information  as  to  current 
arithmetic  practices.  A.  H.  B. . 

W.  A.  A. 


EDITOR'S  INTRODUCTION 

Two  small  boys  were  overheard  in  conversation  as  they 
went  out  from  the  morning  assembly  exercises  of  a  given 
school,  just  after  the  thirteenth  chapter  of  First  Corinthians 
had  been  read.  One's  remark  to  the  other  fairly  expressed 
the  estimate  which  has  been  placed  on  the  value  of  the 
elementary  school  studies.  The  lad's  comment  was,  "  Readin, 
ritin,  and  rithmetic,  and  the  greatest  of  these  is  rithmetic." 
Arithmetic  has  claimed  a  large  part  of  the  school's  time ; 
it  has  presented  the  very  citadel  of  difficulty  to  the  pupil ; 
and  it  has  been  made  the  object  of  first  importance  by  the 
teacher. 

While  the  above  statements  are  true  there  are  other  facts 
in  striking  contrast.  Results  at  present  secured  from  the 
study  of  arithmetic  are  most  unsatisfactory.  Several  inves- 
tigations in  different  parts  of  the  country  have  shown  that 
pupils  go  out  from  the  schools  not  understanding  the  pro- 
cesses of  modern  business,  and  not  able  to  make  trustworthy 
computations.  Schools  are  not  realizing  an  educational 
result  from  arithmetic  commensurate  with  the  time  and 
effort  spent  on  it,  and  the  present  situation  is  one  which 
calls  for  careful  consideration. 

In  the  first  place  arithmetic  is  of  all  the  elementary  school 
studies  the  one  most  in  danger  of  becoming  conventionalized. 
Teacher  and  textbook  tend  to  perpetuate  themselves. 
Marked  changes  may  occur  in  practical  affairs  about  which 
the  teacher  and  the  textbook  author  know  little.  Thus 
it  was  that  the  mercantile  methods  of  the  eighteenth  cen- 


viii  EDITOR'S    INTRODUCTION 

tury  continued  as  the  basis  of  arithmetical  instruction  in  the 
nineteenth  century,  long  after  those  methods  had  disap- 
peared ;  and  thus  there  are  traces  of  the  eighteenth  century 
still  to  be  discerned  in  the  arithmetics  of  the  present.  If 
the  merchant  of  an  earlier  generation  were  to  reappear  and 
attempt  to  do  business,  he  would  be  dumfounded  by  the 
changes  in  commercial  procedure;  but  he  could  scarcely  be 
at  a  greater  disadvantage  than  are  pupils  who  are  trained 
after  the  methods  which  he  had  used  and  then  sent  out  to 
take  their  places  in  the  world  of  to-day. 

The  book  herewith  presented  is  an  honest  attempt  to  set 
forth  correctly  the  fundamental  operations  of  modern  busi- 
ness, and  to  furnish  a  goodly  amount  of  drill  on  the  kind 
of  computations  which  make  up  present-day  commercial 
practice.  Messrs.  Bigelow  and  Arnold  have  spared  no 
pains  to  inform  themselves  on  current  business  operations, 
and  the  editor  believes  that  they  have  presented  their  infor- 
mation succinctly  and  logically  arranged  for  purposes  of 
instruction.  The  book  has  been  the  result  of  much  labor 
in  its  first  preparation,  and  as  first  prepared  it  was  duplicated 
to  serve  as  a  text,  and  modified  in  class  instruction  for 
several  years.  In  addition  to  this  it  has  been  revised  and 
adapted  in  accordance  with  suggestions  of  experienced 
teachers  in  different  parts  of  the  country.  It  is  believed 
that  all  this  has  resulted  in  a  book  of  accurate  information, 
of  sound  mathematical  basis,  and  of  high  teaching  quality. 

Several  features  of  the  book  will  commend  themselves 
to  teachers.  Among  these  are  the  script  illustrations  as 
models  for  trial  balances,  ledger  accounts,  time  sheets, 
accounts  of  sales,  etc.  These  have  been  executed  by  the 
skilful  pen  artist  and  illustrator,  Mr.  E.  C.  Mills. 

Chapter  IV  on  Fractional  Parts  presents  a  natural  and 
easy  approach  to  fractions,  and  will  be  found  of  great  practi- 
cal value.     The  authors  have  had  the  courage  to  put  deci- 


EDITOR'S   INTRODUCTION  ix 

mals  ahead  of  common  fractions,  treating  them  as  they  should 
be  treated,  simply  as  a  descending  scale  in  our  decimal  nota- 
tion, of  which  whole  numbers  are  an  ascending  scale. 

The  book  will  be  found  to  have  little  of  the  impractical 
and  troublesome  G.  C.  D.  and  L.  C.  M.  problems.  Fractions 
of  large  denominations  are  not  introduced,  as  they  present 
difficulties  and  are  almost  never  encountered  except  in  the 
arithmetics.  Square  root  and  the  treatment  of  mensuration 
are  disposed  of  in  connection  with  weights  and  measures. 
The  antiquated  percentage  problems  to  be  solved  by  the  use 
of  formulas  are  omitted,  as  are  the  conventional  percentage 
formulas  themselves.  Partial  payments  is  relegated  to  an 
unimportant  place  in  the  Chapter  on  Interest.  The  so-called 
true  discount  is  eliminated  from  the  textbook  as  it  is  from 
business;  and  partnership  and  proportion  are  so  treated  as 
to  bring  them  within  the  practice  of  the  actual  world. 

In  some  particulars  the  book  is  not  as  revolutionary  as 
the  authors  would  have  desired,  but  it  is  believed  to  be  as 
revolutionary  as  it  could  be  without  breaking  with  the  prac- 
tice of  the  schools.  This  text,  it  is  believed,  will  prove  a 
logical  and  easy  completion  of  the  average  elementary  course 
in  arithmetic.  It  should  find  a  place  in  the  last  years  of  the 
grammar  school,  as  the  finishing  book  of  the  ungraded  school, 
and  for  the  first  high  school  year. 

Throughout,  this  book  will  be  found  to  use  the  method 
of  concrete  presentation.  The  pupil  is  constantly  asked 
to  consider  problems  with  the  thought  of  determining  the 
particular  solution  which  will  get  a  result  the  most  directly. 
Thus  there  is  an  absence  of  any  ".wooden  "  working  by  rule. 
At  every  turn  the  pupil  is  required  to  select  his  solution, 
and  to  use  his  head  in  applying  the  form  selected.  This 
cannot  fail  to  produce  clear  thinking  and  a  facility  in  com- 
putations which  will  give  accuracy.  The  problems  of  easy 
solution  for  mental   arithmetic  offer   one  valuable  feature 


X  EDITOR'S    INTRODUCTION 

of  the  book.     These,  largely  used,  will  develop  power  of 
accurate  thought  and  power  of  expression. 

A  limited  range  of  treatment,  with  freedom  of  explana- 
tion and  plenty  of  drill  on  the  fundamentals ;  not  too  much 
arithmetic  attempted,  but  what  is  attempted  done  well ; 
not  generalized  and  abstract  number,  but  arithmetic  related 
to  the  life  experiences  of  the  child ;  problems  selected  from 
the  world  around  about  the  child ;  not  a  treatment  which 
shall  be  "  milk  for  babes,"  but  one  which  will  afford  such  a 
ruggedness  of  drill  as  will  make  arithmetic  a  means  of  disci- 
plinary education  and  a  book  of  accurate  information,  — 
these  are  the  standards  which  the  "Elements  of  Business 
Arithmetic  "  has  sought  to  meet.  Possibly  it  has  failed  in 
some  particular,  but  it  is  published  in  the  confident  belief 
that  there  is  a  large  place  for  such  a  presentation  as  is  here 
attempted.  The  authors  and  the  editor  have  the  satisfaction 
of  having  worked  long  and  faithfully.  They  invite  correc- 
tions and  suggestions  for  the  improvement  of  the  book. 

C.  A.  H. 


CONTENTS 

CHAPTER  PAGB 

I.     Addition  and  Subtraction 1 

11.     Multiplication  and  Division 16 

III.  Decimals 23 

IV.  Fractional  Parts 32 

V.    Fractions 53 

VI.     Measures  of  Length 67 

VII.    Measures  of  Area .  72 

VIII.     Measures  of  Volume 109 

IX.    Measures  of  Time 126 

X.  Measures  of  Weight      .        .        .        .        .        .        .  137 

XL    Measures  of  Value 141 

XIL    French  Metrical  System 149 

XIII.  Percentage 157 

XIV.  Trade  Discount 172 

XV.    Commission 180 

XVL     Taxes  and  Duties 185 

XVII.     Interest 192 

XVIII.    Banking  and  Discount 205 

XIX.    Stocks  and  Bonds 221 

XX.    Insurance .        .        .  231 

XXI.     Proportion 240 

XXII.  Proportional  Parts  and  Partnership        .        . '       .  250 

Index 255 


XI 


THE  ELEMENTS  OF  BUSINESS  AEITHMETIC 


ADDITION  AND  SUBTRACTION 

1.  Combinations  to  9.  Pupils  who  use  this  book  will 
probably  know  the  combinations  of  numbers  up  to  9,  but  it 
may  be  presumed  that  they  are  not  sufficiently  quick  and 
accurate  in  the  use  of  these  combinations.  Thorough  drills 
should  be  given  and  continued  until  pupils  can  think  groups 
as  a  whole  and  can  instantly  name  sums  without  naming 
the  parts. 

Note.  —  There  are  but  forty-five  possible  combinations  of  numbers 
up  to  9.  The  common,  device  of  having  each  one  of  the  combinations 
on  a  separate  card  and  giving  results  at  sight  rapidly  and  in  varied  or- 
der, forms  a  practical  drill.  Each  card  may  have  the  same  combina- 
tion on  its  back  with  the  order  of  the  figures  reversed. 

DRILL   TABLE 

The  forty-five  two-figure  combinations.     Name  sums  at  sight. 
74241343314221189856455 


7 

6 

5 

3 

7 

6 

2 

3 

2 

5 

1 

2 

1 

3 

1 

9 

9 

8 

5 

1 

4 

3    4 

7 

1 

5 

6 

6 

8 

9 

8 

7 

7 

4 

9 

7 

6 

7 

5 

3 

2 

4 

5 

7 

6 

2 

8 

6 

6 

9 

6 

1 

2 

3 

5 

8 

3 

8 

7 

9 

9 

8 

9 

9 

8 

4 

2 

2.  Combinations  to  19.  The  following  combinations 
should  be  drilled  upon  and  learned  in  the  same  way  as  those 
in  Sec.  1.     They  are  important  for  rapid  addition  (Sec.  3). 

B  1 


2.  ELEMENTS  OF  BUSINESS  ARITHMETIC 

The  instant  calling  of  results  should  be  insisted  upon.     Drill 
cards  as  above  suggested  will  be  useful. 

Note.  —  In  making  these  combinations,  the  second  number  should  be 
thought  of  as  one  ten  and  a  given  number  of  units  and  not  as  a  given 
number  of  units  and  one  ten.  Thus,  in  combining  14  and  13  we  should 
think  of  24  and  3  ;  in  combining  17  and  18  of  27  and  8. 

DRILL  TABLE 

Forty-five  more  combinations,  11  to  19.     Name  sums  at  sight. 

12     13     11     17     11     17     12     15     12     16     19     17     11     14    15    12    19 
15    18    11     17     16     14     12     16     13     13     14    13     14    15     17    11    16 


15 

18 

17 

12 

16 

15 

18 

17 

13 

16 

11 

19 

14 

19 

16 

19 

13 

15 

18 

16 

17 

12 

13 

16 

18 

11 

16 

19 

18 

13 

15 

14 

19 

JL3 

14     14     18     18     19     12     18     17    17     12     11 
1418121113141511191915 

3.  Adding  by  Groups.  In  adding  columns  of  figures  one 
should  think  first  of  the  sum  of  each  group,  then  group  the 

^  .^       sums  and  name  the  sum  of  the  groups.     In  the 

o  problem  given,  think  and  name  the  sum  of  the 

-.  07      fii^st  group  as  14  without  naming  the  figures  of 

o  the  group,  the  sum  of  the   second  group  as  13, 

^  2_      then  the  sum  of  13  and  14  is  thought  and  named. 

r  Seeing  the  next  group  as  10,  we  think  the  sum  of 

Q w  .       27  and  10,  saying  37;   then,  11,  48.      In  adding, 

r  name  the  results  only,  as  :  14,  27,  37,  48. 

^  Note. — Double  drill  may  be  had  from  the  same  problems 

by  adding  both  up  and  down;  e.g.  11,  21,  34,  48.  This  is 
also  a  way  of  detecting  errors.  If  the  same  result  is  obtained  from 
adding  both  ways,  the  sum  is  probably  correct. 

4.  Adding  Two  Columns.  By  practice,  two  columns  of 
figures  may  be  readily  added  at  once.  The  tens  should  be 
combined  first  and  the  units  added  to  their  sum  (Sec.  3). 


ADDITION  AND  SUBTRACTION  3 

The   problems  below  may  be  used  for  practice  in  both 

single  and  double  column  adding.     Add  both  up  and  down. 
Drill  until  the  group  sums  can  be  named  quickly. 

Add. 

37  95  67  68  56  45  27  35  46  24  99  48  71  89  76  32  33  45 

40  36  76  66  75  72  88  87  72  98  76  79  88  15  90  54  38  98 

63  36  76  97  57  83  33  77  32  68  71  90  13  31  16  58  84  17 

84  66  54  78  86  96  38  55  33  19  10  34  42  34  86  92  10  57 

67  88  38  86  84  48  47  94  65  56  25  56  75  85  83  17  42  56 

22  71  95  84  32  -23  47  85  65  76  81  34  56  87  34  50  60  70 

71  95  84  32  23  47  85  65  76  81   8  78  19  30  54  76  58  43 

45  24  98  76  67  19  14  67  32  78  76  14  98  88  28  34  13  65 


Practice  adding  by  groups. 

1.   2.   3.   4.   5.   6.    7.    8.    9.   10.   11.   12.   13. 

180  718  678  432  178  852  458  37  458  123  7689  23  1768 

717  678  743  327  187  918  1079  1876  7  7453  4  717  324 

324  187  509  834  778  415  2345  324  6789  1064  318  6324  5324 

873  548  791  130  543  915  718  98  4317  8796  1745  6  1098 

376  876  987  734  435  478  3458  1756  5692  187  678  1  367 

324  578  713  105  791  698  1234  304  2934  67  5487  75  84 

943  528  281  375  905  768  5432  32  701  5432  34  150  69 

473  432  958  104  548  785  7981  7615  45  478  91  235  75 

432  958  104  548  785  442  2701  89  6194  91  3605  768  150 

573  473  109  423  478  871  __71  _432  5061  2398  _148  _^  1785 

Add  across.  Add' down.  Pfoblems  like  these  are  useful  in  training 
to  accuracy,  and  frequently  occur  in  offices.  If  the  final  results  are  not 
alike,  the  student  should  find  and  correct  mistakes  without  awaiting 
correction  by  the  the  teacher. 


1.  324+625  +  463  +  526  =  ?/-  2.  103  +  184  +  216  +  135  +  320  =  ? 
462  +  375  +  836  +  472  =  ?  '  413  +  204  +  311  +  401  +  405  =  ? 
378  +  643  +  728  +  645  =  ?  ■      764  +  835  +  735  +  631  +  987  =  ? 

643  +  372  +  654  +  528  =  ?  367  +  543  +  263  +  413  +  187  =  ? 

584  +  653  +  936  +  364  =  ?  v_^^_  185  +  371  +  213  +  715  +  478  =  ? 
?+?    +   ?    +    ?=?    \vi\  \    ?+?+?    +    ?    +    ?=? 


4  ELEMENTS  OF  BUSINESS  ARITHMETIC 

3.  265  +  167  +  324  +  734  +  178  =  ?  '     ' 

562  +  713  +  432  +  817  +  384  =  ?   . 
276  +  810  +  305  +  978  +  308  =  ?  , 

417  +  187  +  523  +  734  +  198  =  ?  .icSl 

672  +  432  +  278  +  598  +  471  =  ?  Tgj^ — ■ 

?+?    +    ?    +    ?    +    ?=?      '    -'^^ 

"\1*^  .    :       -  -  '  '^'' 

5,   Adding  Long  Columns.      Civil  Service  Method.     In  add- 
ing long  columns  of  figures  it  is  often  desirable  to  retain 
$791.52     *^®  exact  sum  of  each  column.     Errors  are  more 
604.83     easily  located  and  unnecessary  re-adding  of  col- 
879.26      umns  which  are  correct  is  often  avoided.     Gen- 
243.79     erally  speaking,  this  practice  should  always  be 
732.46      used  for  columns  of  more  than  ten  numbers. 
47.95  This  is  sometimes  known  as  the  "  Civil  Service 

856.43  Method,"  presumably  from  its  large  use  in  gov- 
497.65  ernment  offices.  It  may  be  frequently  used  to 
541.26  advantage  in  many  offices.  Practice  using  this 
616.72      iiiethod  in  the  problems  given  after  Sec.  6. 

857.94  Note. — Drill  in  urriting,  from  dictation,  long  columns 

^  of  numbers  of  varied  size,  so  that  neat  vertical  columns 

Q  g  will  be  secured.     Verify  answers  by  adding  downward,  if 

ro  the  first  addition  was  upward,  or  vice  versa. 

^^  6.    Addition    Proof    by   Check.     In    counting- 

rooms  or  elsewhere  where  long  columns  of  fig- 
qt)  0009.81  ^j,gg  ^j.Q  common,  some  system  for  checking  the 
correctness  of  results  of  addition  is  often  resorted  to.  Such 
devices  are  of  various  kinds,  but  all  consist  of  some  variation 
of  a  casting  out  process,  as  of  the  9's  or  ll's. 

The  Unitate  method  is  here  presented.  It  consists  of  re- 
moving all  the  nines  from  the  numbers  added  and  from  the 
sum.  The  remainder  would  necessarily  be  less  than  nine 
and  being  a  single  figure  is  termed  a  unitate.  When  the 
unitate  of  the  numbers  added  is  the  same  as  the  unitate  of  the 
8um^  the  addition  is  usually  correct. 


60 


ADDITION  AND  SUBTRACTION  5 

The  process  of  finding  the  unitate  of  a  given  number  or, 
as  it  is  often  called,  "  casting  out  the  nines,"  is  based  on  the 
facts  that  our  system  of  notation  is  a  decimal  one  and  that 
there  is  but  a  single  unit  of  difference  between  9  and  the 
basis  of  our  notation,  10. 

Every  digit  in  our  notation  stands  for  the  number  of  tens 
(or  powers  of  ten)  represented  by  the  digit.  Now  there  are 
as  many  nines  in  each  of  these  as  there  are  tens  and  there 
would  be  a  remainder  of  a  single  unit  from  each  ten.  But 
as  each  digit  represents  the  number  of  tens,  there  would  be 
a  number  of  units  equal  to  the  digit  itself  remaining,  after 
all  the  nines  had  been  taken  out.  The  sum  of  the  digits  of 
any  number  would  represent,  then,  the  number  of  units 
remaining  after  all  the  nines  had  been  "cast  out."  If  this 
sum  is  a  number  with  more  than  one  digit,  it  still  contains  a 
nine  and  the  sum  of  its  digits  would  be  the  true  remainder. 
When  this  remainder  consists  of  but  one  digit,  it  is  the 
desired  unitate. 

To  find  the  unitate  of  87,564,892,  add  the  digits  and  their 
sum  is  49  ;  adding  these  gives  13,  and  adding  these  gives  4, 
the  unitate. 

Unitates. 

Thus,  32,543 8 

16,789 3     h'^i 

11,942 2      :  \ 

27,683 1 

38,578 5 

96,541 3 

73,847 5 

297,923 5 

In  theory,  the  unitates  of  every  addend  are  themselves 
reduced  to  a  unitate,  which  should  equal  the  unitate  of  the 
correct  answer. 


6 


ELEMENTS  OF  BUSINESS  ARITHMETIC 


In  practice,  the  "unitate  of  the  first  addend  (8)  is  written 
and  then  added  to  the  digits  of  the  second  addend  and  the 
whole  reduced  to  its  unitate  (3).  This  is  again  added  to 
the  digits  of  the  third  addend  and  the  third  unitate  is  found 
(2).  This  process  is  continued  until  the  unitate  of  the  last 
addend  is  found,  which  should  be  the  same  as  the  unitate  of 
the  answer. 

Checking  by  the  unitate  is  not  an  absolute  proof.  It  will 
not  detect  an  error  in  transposition  of  digits,  or  where  the 
wrong  digits  total  the  same  as  the  right  ones.  It  is  a  satis- 
factory check,  however,  in  most  cases. 


PROBLEMS 

Solve  and  prove  the  following : 
1. 


-^y^^-t^t^a^  yOi2^<?^iz'<?'Z^^:-.^<t^i^^^,.z^ /, /(? — 

Cre«i 

/ 

{^  (o-x^Z/ZJ/-.  ~^L'ff-^2A..<^6-to-r' 

3  0^Z 

7' 

i. 

/^,^yL-C'A-€i.-'7^i.'tyU..d.-^^ 

za./  ^ 

ZO 

3 

/^«^-Zt£,,i>  7^.^u:-f>^^-^^-^^^^ 

/  /^J 

S-C 

€^. 

^7^-rz.<^<?C^ 

U^ 

^0 

^77 

o  o 

oT 

-^5^£^^^>^^^.^^^#/,  ^  ^;fid..^;z^^ 

3yo 

o  o 

CT 

-'^^^*^4^'-5f^t-^>*2'/-^/,  Jl  (o  ^y^o^'^-e^ri^ 

7^ 

O  0 

^ 

(o^^t^^z-c^n^j..^^ 

f^ 

0  o 

^ 

v^^W-^ii'i.^-i^^  ^z-^i^  o^2-<ui^><^-«i--*z-^ 

JO 

o  o 

// 

-^,  ^  -e^^'tz.^yt-^ 

2^7 

o  o 

r 

X  2S 

0  0 

rus 

0  0 

JOZ 

0  o 
0  o 

/f 

OT^^-^Oa^^ 

J  0  0 

0  0 

2  0 

/^.,<yt^<>A^fiyy2^  ^^d^^c^^b^^b^ 

33/ 

J-0 

ZO 

C/ '^ -J^'-Cl^<.A'<'t^-it^cS 

779 

0  0 

2.0 

uC-i^z-.^l^-.^r-T^L.^c^^iz^'^''^^ 

^Ci' 

0  0 

ADDITION  AND  SUBTRACTION 


BAUSCH  &  LOME  OPTICAL  COMPANY 

MANUFACTURERS 


^ 


'.-^^^-r-^^, 


/  ^^,.>&^^r^.^^^.J^j'U^  - 


,i'^^.;>'4--r?--i^!--^::J>^^.^ -^^  .  y^v.^^?^-^ -i^^^-X_  /*■" 


^ 


2.'^^  'f^.J'/^^  ^^J^^^.  ^^^^  f- 


-^ 


tU^. 


>-v!^-/  ■^A^^^^ 


J-^-^L^  t-y6^^/Ce 


■C-1H,'Tn.-f2.r^ - 


'(yOlJT^y},^^^^  y^f^ 


-///t^^y!. 


'^.■'>^-^^^'y?^y 


-^k^X>. 


^a^.A 


^.^^.^yiy~k?<-t<^^^^ 


■£:Pf^^^,:?t/y. 


JM.  >^^^^-.^^^.^^ 


^ 


^■<^n=^^^    /2.rr^y.-/^J. 


r-.,^.. 


,^^^f 


??^^f?r^ 


Th'yf^r^ 


^(0^ 


r^^^y^i 


^Af. 


/o.^  ^^^^y/^'y>^. 


<^^M,i^^ 


/d^'A^''^^-^^-^'^^^<^-p^ 


Grand  Rapids,  Mich.. l/{..^2^^.^  Zt^, — 19. 

The  Hancock  Furniture  Company 


356-359  State  Street 


i^ 


.^Skz 


■^^^..^y^^.^f7-y7. 


^~^^t 


^<?^V-Z^>;^7^   (jJJy^ 


^.-fT-^ 


-/^J-^A^aA- 


Ajl 


.^2^i-^e^i^^£i.=:^ 


/Z 


>r7.^^^^ 


/  ^ 


=^X-^.^r^1^. 


'^..<^^  /^^.^d-J^ 


J^ 


.^ 


."ri^'^:^^':CrtC^..<y-t;^gg^^  . 


-^.^-^^ 


>^s 


id2;l 


8  ELEMENTS  OF  BUSINESS  ARITHMETIC 

4.   List  of  appropriations  by  Congress,  for  bienniuni. 


19_ 

19_ 

Deficiencies 

$  19,651,968.25 

^  25,083,395.78 

Legislative,  Executive,  and  Judicial 

27,598,653.66 

28,558,258.22 

Sundry  Civil 

61,763,709.11 

49,968,011.34 

Support  of  the  Army 

. 

77,888,752.83 

77,070,300.88 

Naval  Service    . 

81,876,791.43 

97,505,140.94 

Indian  Service   . 

8,540,406.77 

9,447,961.40 

Rivers  and  Harbors  . 

20,228,150.99 

10,872,200.00 

Forts  and  Fortifications 

7,188,416.22 

7,518,192.00 

Military  Academy     . 

652,748.67 

973,947.26 

Pensions     . 

139,847,600.00 

138,360,700.00 

Consular  and  Diplomatic 

1,968,250.69 

2,020,100.69 

Agricultural  Department 

5,978,160.00 

5,902,040.00 

District  of  Columbia 

8,638,097,00 

11,018,540.00 

Miscellaneous     . 

3,025,064.95 

2,860,828.52 

Total 

5.    The  cotton  crop  of  the  United  States  by  states  for  five  years. 


States 

1 

2 

3 

4 

5 

North  Carolina 

480,000 

400,000 

426,000 

504,000 

400,000 

South  Carolina 

960,000 

874,000 

948,000 

955,000 

845,000 

Georgia 

1,448,000 

1,226,000 

1,493,000 

1,498,000 

1,405,000 

Florida 

54,000 

57,000 

56,000 

60,000 

55,000 

Alabama 

1,161,000 

1,136,000 

1,287,000 

1,065,000 

1,040,000 

Mississippi 

1,776,000 

1,349,000 

1,460,000 

1,418,000 

1,385,000 

Louisiana 

577,000 

651,000 

851,000 

864,000 

832,000 

Texas 

3,143,000 

2,575,000 

2,682,000 

2,575,000 

2,446,000 

Arkansas 

921,000 

665,000 

771,000 

938,000 

855,000 

Tennessee 

381,000 

240,000 

229,000 

303,000 

255,000 

All  others 

334,000 

267,000 

498,000 

578,000 

516,000 

Total  crop 

r    ■  .    •; 

v^Jioi.c 

'V  ' 

""'  ■.   '  'i 

'■ 

ADDITION  AND  SUBTRACTION 
6.    Add  down.     Add  across. 


7654 

4753 

1954 

1763 

8197 

6548 

? 

1458 

3674 

1756 

3263 

3954 

3245 

? 

5786 

9876 

1753 

5132 

6587 

3642 

? 

4327 

9876 

587 

3674 

5743 

6798 

? 

56 

7815 

2301 

1567 

4326 

189 

? 

1587 

9817 

3246 

1685 

8542 

7432 

? 

1563 

6743 

9816 

1076 

875 

1904 

? 

8543 

3425 

178 

25 

3607 

98 

? 

5674 

8795 

5432 

5768 

5843 

2345 

? 

+ 


7.    Departmental  sales  for  the  week  ending  June  15,  19 


Days 

Clothing 

Dry  Goods 

Furnish- 
ings 

Millinery 

Groceries 

Total 

Monday 

695.50 

894.30 

175.65 

325.45 

678.10 

? 

Tuesday 

546.15 

716.98 

243.25 

817.42 

313.48 

? 

Wednesday 

981.76 

654.32 

145.60 

567.89 

543.26 

? 

Thursday 

578.90 

765.10 

324.65 

687.58 

987.60 

? 

Friday 

842.45 

918.75 

216.40 

561.46 

674.15 

? 

Saturday 

985.50 

818.40 

456.12 

764.55 

925.48 

? 

Total 

? 

9 

? 

? 

? 

? 

8.  The  records  of  a  post-office  show  the  following  mail  for  six  days : 
Monday,  registered  letters,  625;  ordinary  letters,  14,570;  postal  cards, 
2134;  book  packets,  957;  parcels,  184;  newspapers,  25,514.  Tuesday, 
registered  letters,  541 ;  ordinary  letters,  13,576  ;  postal  cards,  2134  ;  book 
packets,  587;  parcels,  146;  newspapers,  26,156.  Wednesday,  registered 
letters,  750;  ordinary  letters,  14,569;  postal  cards,  3456 ;  book  packets, 
1056;  parcels,  178;  newspapers,  24,356.  Thursday,  registered  letters, 
587  ;  ordinary  letters,  13,452 ;  postal  cards,  2451 ;  parcels,  143 ;  news- 
papers, 23,781.  Friday,  registered  letters,  547;  ordinary  letters,  13,567; 
postal  cards,  1346 ;  book  packets,  890 ;  parcels,  157 ;  newspapers,  26,543. 


10  ELEMENTS  OF  BUSINESS  ARITHMETIC 

Saturday,  registered  letters,  857 ;  ordinary  letters,  15,472 ;  postal  cards, 
3145  ;  book  packets,  789 ;  parcels,  245  ;  newspapers,  23,100. 

Arrange  these  facts  in  tabular  form,  in  six  columns,  under  proper 
headings.  Find  the  total  number  of  pieces  of  mail  for  each  day,  the 
total  number  of  pieces  of  each  class,  and  the  total  number  for  six  days. 

7.  Method  of  Subtraction.  Whether  the  pupil  subtracts 
by  the  method  of  borrowing  from  the  minuend  or  by 
adding  to  the  subtrahend,  is  immaterial.  Ordinarily  he 
should  be  permitted  to  use  the  method  previously  taught 
him.  The  effort  should  be  toward  facility,  which  may  only 
be  acquired  by  practice.  The  examples  here  given  are  sug- 
gestive rather  than  sufficient.  If  when  these  are  completed 
there  is  not  the  power  of  quick  and  accurate  subtraction, 
much  additional  work  should  be  given. 

If  another  method  is  desired,  that  known  as  the  addition 
method  is  suggested.  It  has  the  advantage  of  using  the 
addition  combinations  previously  taught  and  involves  no 
new  process. 

In  the  accompanying  problem,  we  say  4  and  5  are  9,  writing 
.     the  5.     Then,  8  and  8  are  16,  writing  the  8  and  carrying  the  1 
■  as  in  addition.     Then,  10  (9  +  1)  and  0  are  10,  writing  0 ;  and  1 

(carried  from  10)  and  1  are  2,  writing  1. 

8.  Making  Change.  The  addition  method  of  making 
change  is  the  one  most  generally  used  by  tellers,  and  is  un- 
questionably the  most  accurate.  Thus,  a  ten-dollar  bill  is 
tendered  in  payment  of  a  bill  of  $2.85.  Counting  out  suc- 
cessively a  five-cent  piece,  a  dime,  two  dollars,  and  a  five- 
dollar  bill,  the  sums  are  named  as  follows:  $2.85,  $2.90, 
$3,  $4,  $5,  $10. 

Name  the  coins  and  bills,  and  the  amount  of  change  to  be 
given  in  each  of  the  following  : 

1.  From  $1,  for  a  bill  of:  17^,  43^,  65/-,  72;^,  28 J^,  10 ^  15^,  40^, 
35j^,  87/^. 


ADDITION  AND  SUBTRACTION  11 

2.  From  |2,  for  a  bill  of:  |1.26,  $  1.40,  $1.47,  $1.61,  $1.75,  $1.55. 
$1.69,  $1.83,  $1.19,  $1.33. 

3.  From  $.5,  for  a  bill  of:  $3.50,  $1.12,  $2.32,  $3.37,  $2.87,  $0.79, 
$4.11,  $1.78,  $3.56,  $2.75,  $4.15. 

4.  From  $10,  for  a  bill  of:  $3.50,  $4.75,  $6.32,  $7.28,  $4.87,  $9.15, 
$8.5.5,  $6.70,  $1.95,  $4.28. 

5.  From  $20,  for  a  bill  of:  $17.35,  $14.32,  $13.45,  $10.75,  $9.15, 
$  3.85,  $  12.10,  $  14.30,  $  8.24. 

9.  Horizontal  Subtraction.  Subtract  the  following  with- 
out rearranging.  Find'  the  sum  of  the  minuends,  the  sum 
of  the  subtrahends,  and  the  sum  of  the  remainders. 

1.   3,264,873-  286,729  =  ?                   3.       $867.50-  $742.25  =  ? 

328,629-  124,962=?  329.87-  124.68  =  ? 

729,687  -  638,469  =  ?  1768.42  -  938.89  =  ? 

382,962-  146,702  =  ?  2762.48-  1262.34-? 

2,678,212  -  1,476,388  =  ?  3786.32  -  439.37  =  ? 

729,326-  384,578  =  ?  9623.29-  3674.28  =  ? 

368,742-  176,386  =  ?  9627.42-  7672.91=? 

504,726  -  386,275  =  ?  1076.16  -  729.78  =  ? 


?        -         ?         =? 

2.   496,827  -  268,794  =  ? 

1,986,702  -  1,346,825  =  ? 

2,787,543  -  1,968,729  =  ? 

79,843  -   67,983  =  ? 

4,869,625  -  2,278,631  =  ? 

967,198  -     798,631  =  ? 

472,398  -     126,535  =  ? 

848,716-    432,567  =  ? 

V       -       ?       r?  ?       _       ?       =? 

Find  the  new  balances  in  the  following  banking  individual 
ledger  accounts  by  adding  the  deposits  to  the  balances  and 
subtracting  the  checks.  Find  the  total  of  balances,  checks, 
and  deposits.     Prove. 


?  - 

? 

=  ? 

$6487.50- 

$3228.25: 

=  ? 

329.75  - 

128.95 : 

=  ? 

6756.50  - 

2278.36  : 

=  ? 

4798.60  - 

3128.75 : 

=  ? 

12986.72  - 

9647.22 : 

=  ? 

3678.45  - 

2968.42  : 

=  ? 

728.32  - 

648.25 : 

=  ? 

2198.65  - 

1297.70 : 

=  ? 

12 


ELEMENTS  OF  BUSINESS  ARITHMETIC 


1. 


Names 

Balances 

Checks 

Deposits 

Balances 

Ames   Wm   E 

865.52 
584.32 
954.60 
523.40 
976.35 
126.65 
925.43 
1214.34 
752.30 
178.95 
986.57 

321.56 
127.55 
532.58 
305.45 
532.78 

38.72 
413.86 
615.47 
456.87 

34.56 
485.74 

675.80 
220.00 
276.50 
560.00 
125.40 
423.80 
575.94 
213.44 
435.87 
698.00 
325.34 

Bentley,  C.  A. 

Dayton,  F.  R. 



Frank  G  A 

G-ramrn    D   C 

Hughes,  C.  M. 
Innes  U.  P 

Justin,  John 
King,  A.  S. 

? 

? 

? 

?. 

Names 

Balances 

Checks 
IN  Detail 

Total  Checks 

Deposits 

Balances 

Anson,  E.  M. 

5671.80 
1544.42 
2345.60 
5467.80 

967.85 
1267.98 

845.34 
3289.07 

876.35 

125.50 

232.20 

678.90 
750.00 
142.76 
525.00 
453.50 

134.75 

1100.00 
190.00 
253.78 

1234.56 
26.78 

325.40 

546.70 

546.70 
756.05 
854.32 
768.90 
987.45 
1200.00 
584.32 
542.08 

Barnes,  T.  D. 

Cowles,  E.  M. 

Doyle,  F.  E. 

Farish  Bros. 

Grim,  G.  L. 

Haines,  F.  R. 

Johns,  B.  I. 

Love,  P.  V. 

? 

V 

? 

? 

? 

ADDITION  AND  SUBTRACTION  li' 

10.  Problems  for  Explanation.  In  problems  where  the  pro- 
cesses themselves  are  very  simple,  the  more  difficult  work  of 
accurate  expression  in  equation  form  and  of  explanation 
should  be  carefully  taught.  Exactness  in  language  is  the 
only  proper  expression  for  an  exact  science. 

A  suggestive  statement  and  explanation  are  given.  Set 
forms  of  explanation  are  not  desirable,  but  a  correct  use  of 
mathematical  symbols  and  of  clear  and  accurate  English  in 
the  explanation  of  a  problem  should  be  insisted  upon.  Use 
the  fewest  words  possible  to  express  the  thought  clearly. 

Problem.  —  During  one  season  a  jobbing  carpenter  built 
five  dwellings  which  cost  him  respectively  13176,  $5194, 
$1342,  16950,  and  $788.  He  received  for  building  them 
$3875,  $6820,  $1280,  $7896,  and  $875.  What  were  his 
season's  profits  ? 

$3176  +  $5194  +  $1342  +  $6950  +  $788  =  $17,450. 
$3875  +  $6820  +  $1280  +  $7896  +  $875  =  $20,746. 
$20,746  -  $17,450  =  $3296,  season's  profits. 

Explanation.  —  The  total  cost  of  building  the  five  build- 
ings is  the  sum  of  $3176,  $5194,  $1342,  $6950,  and  $788,  or 
$17,450.     The  total  amount  received  is  the  sum  of  $3875, 

$6820,  $1280,  $7896,  and  $875,  or  $20,746.     His   profits, 
therefore,  are  the  difference  between  $20,746  and  $17,450,  or 

$3296. 

PROBLEMS 

1.  A  grain  dealer  bought  15,640  bushels  of  wheat,  and  sold  at  one  time 
3465  bushels,  at  another  time  4205  bushels,  and  at  another  time  1080 
bushels.     How  many  bushels  remained? 

2.  A  man  deposited  in  a  bank  .|9672.  He  drew  out  at  one  time 
$4234,  at  another  $1700,  at  another  $762,  and  at  another  $49.  How 
much  remained? 


14  ELEMENTS  OF  BUSINESS  ARITHMETIC 

3.  A  teacher's  salary  was  $1200.  His  living  expenses  were  $760. 
He  paid  $314  for  a  lot  and  $95  for  a  horse.  How  much  of  his  salary- 
remained  ? 

4.  A  merchant  in  a  year  bought  goods  to  the  amount  of  $  8750.  He 
paid  for  clerk  hire  $6735,  and  for  rent  $318.  For  how  much  must  he 
sell  his  goods  in  order  to  clear  $  1250  ? 

5.  I  sold  a  farm  for  $  9625  and  a  house  for  $  3275.  Lost  $  475  on  the 
farm  and  gained  $  360  on  the  house.  What  did  each  cost  me  ?  What 
was  the  total  gain  or  loss  ? 

6.  The  population  of  Indiana  in  1900  was  2,516,462.  The  population 
of  the  principal  cities  of  the  state  was  as  follows :  Indianapolis,  169,164 ; 
Evansville,  59,007;  Fort  Wayne,  45,115;  Terre  Haute,  36,673;  South 
Bend,  35,999.  How  much  did  the  population  of  the  state  exceed  that  of 
these  cities  ? 

7.  The  distance  from  Chicago  to  Buffalo  is  523  miles,  and  from 
Chicago  to  New  York  980  miles.     How  far  is  Buffalo  from  New  York? 

8.  At  a  sawmill  120,000  feet  of  pine  lumber  were  sawed  in  a  month. 
47,250  feet  of  it  were  sold  to  one  man  and  32,575  to  another.  How  much 
of  the  month's  output  remained  ? 

9.  The  imports  of  sugar  and  molasses  into  the  U.S.  in  one  year 
amounted  to  $  108,387,388,  and  ten  years  later  to  $  101,100,000.  What 
was  the  amount  of  decrease  ? 

10.  Three  persons  bought  a  hotel  valued  at  $45,675.  The  first  agreed 
to  pay  $  8575,  the  second  twice  as  much  as  the  first,  and  the  third  the 
remainder.     How  much  was  the  third  to  pay? 

11.  Borrowed  of  a  bank  at  one  time  $  875,  at  another  $  385,  and  at  an- 
other $ 528.     Having  paid  $  1275,  how  much  do  I  owe? 

12.  A  minister  had  his  life  insured  for  $5000.  At  the  time  of  his 
death,  $375  of  his  salary  was  unpaid;  he  owned  a  farm  worth  $4675, 
but  upon  it  was  a  mortgage  of  $  2385,  and  his  small  debts  amounted  to 
$  879.     What  was  the  value  of  his  estate  ? 

13.  A  stock  dealer  bought  789  cattle  from  A,  and  1249  from  B.  He 
then  sold  228  to  C,  468  to  D,  and  the  remainder  to  E.  How  many  did  E 
buy? 

14.  A  merchant  commenced  business  with  $7500.  The  first  year  he 
gained  $1275,  the  second  year  he  lost  $2475,  the  third  year  he  gained 
$978,  and  the  fourth  year  lost  $674.  How  much  had  he  left  at  the  end 
of  the  fourth  year  ? 


ADDITION  AND  SUBTRACTION 


15 


15.  A  bank  had  $422,785  on  hand.  During  the  day  they  received  on 
deposit  .|  14,657,  and  paid  out  by  check  f  24,570.  How  much  remained 
on  hand  at  the  close  of  the  day  ? 

16.  The  cost  of  my  house  and  lot  was  $  12,860.  I  expended  1 1367 
for  carpenter  work,  $  567  for  bricklaying,  1 6850  for  plumbing,  $  587  for 
painting,  and  $369  for  sodding  and  fencing  the  grounds.  I  then  sold 
the  property  at  a  loss  of  f  135,  receiving  $  7850  in  cash  and  a  note  for 
the  balance.     What  was  the  face  of  the  note? 

17.  During  five  years  a  firm  gained  $36,750.  The  first  year  they 
gained  $7565;  the  second  $4125;  the  third  as  much  as  both  the  first 
and  second  years ;  and  the  fourth  year  the  difference  between  the  gains 
of  the  first  and  second  years.     How  much  did  they  gain  the  fifth  year? 

18.  The  exports  of  cattle  from  the  United  States  during  a  period  of 
nine  years  were  as  follows  :  $159,179;  $439,987;  $1,103,095;  $13,344,195; 
$14,304,103;  $12,906,693;  $31,161,131;  $30,445,249;  and  $39,099,095. 
During  a  later  period  of  seven  years;  '  $23,032,428;  $33,461,022; 
$30,603,796;  $34,560,672;  $36,357,451;  $37,827,500;  and  $30,516,833. 
For  which  period  were  the  exports  greater,  and  how  much  ? 

19.  The  expenditures  for  schools  during  one  year  in  Alabama  were 
$1,583,250;  in  Arizona,  $377,253;  in  Arkansas,  $1,396,594;  in  Cali- 
fornia, $6,401,439;  and  in  Illinois,  $18,167,219.  How  much  more  was 
expended  in  Illinois  than  in  the  other  states  mentioned  ? 

20.    Find  the  balance  of  the  following :  — 


Svcit  ^tional  JSanfe, 


Sa}>toti.  0. 


.^>^^ 


J'^^ 


/- 


.Jyy^^T^n^^^ 


■^^ 


yLyJx^fy^^J- 


Ul. 


-^^-r>-/t 


/  ^rA 


Ul 


sC/>./7^rr-H^^ 


A-2^.,-?^ 


^^~i 


,J'>ird  Ink) 


II 

MULTIPLICATION   AND   DIVISION 
11.    Reference  and  Drill  Table. 


12  3  4 

5 

6 

7   8   9  10 

11  12  13  14  16 

16  17  18  19  20 

2  4  6  8 

10 

12 

14  16  18  20 

22  24  26  28  30 

32  34  36  38  40 

3  6  912 

15 

18 

21  24  27  30 

33  36  39  42  45 

48  51  54  57  60 

4  8  12  16 

20 

24 

28  32  36  40 

44  48  52  56  60 

64  68  72  76  80 

5  10  15  20 

25 

30 

35  40  45  50 

55  60  65  70  76 

80  85  90  95100 

6  12  18  24 

30 

36 

42  48  54  60 

66    72  78  84  90 

96  102  108  114  120 

7  14  21  28 

35 

42 

49  56  63  70 

77  84  91  98  105 

112  119  126  133  140 

8  16  24  32 

40 

48 

56  64  72  80 

88  96  104  112  120 

128  136  144  152  160 

9  18  27  36 

45 

54 

63  72  81  90 

99  108  117  126  135 

144  153  162  171  180 

10  20  30  40 

50 

60 

70  80  90  100 

110  120  130  140  150 

160  170  180  190  200 
176  187  198  209  220 

11  22  33  44 

■^ 

■ST 

77  88  99  110 

121  132  143  154  165 

12  24  36  48 

60 

72 

84  96  108  120  132  144  166  168  180 

192  204  216  228  240 

13  26  39  52 

65 

78 

91  104  117  130  143  156  169  182  195 

208  221  234  247  260 

14  28  42  56 

70 

84 

98  112  126  140  154  168  182  196  210 

224  238  252  266  280 

15  30  45  60 

75 

90  105  120  135  150  165  180  195  210  225| 

240  255  270  285  300 
266  272  288  304  320 

16  32  48  64 

'W 

■w 

112  128  144  160  176  192  208  224  240 

17  34  51  68 

85  102  119  136  152  168  187  204  221  238  255  272  289  306  323  340 

18  36  54  72 

90  108  126  144  162  180  198  216  234  252  270  288  306  324  342  360 

19  38  57  76 

95  114  133  152  171  190  209  228  247  266  285  304  323  342  361  380 

20  40  60  80  100  120  140  160  180  200  220  240  260  280  300  320  340  360  380  400 

12  3  4 

5 

6 

7   8   9  10  11  12  13  14  15  16  17  18  19  20 

12.  Suggestions  for  Study.  It  is  presumed  that  students 
pursuing  this  course  already  know  the  multiplication  tables 
through  the  12's.  It  will  prove  of  great  practical  value  to 
know  them  well  through  the  20's.  If  undertaken,  the  pupil 
should  build  his  own  tables  and  learn  them.     Drill  on  them 

16 


MULTIPLICATION  AND  DIVISION  17 

should  not  stop  short  of  absolute  mastery.     A  half-known 
table  of  15's  will  seldom  be  used. 

Note.  —  Drill  cards  of  convenient  size,  with  the  factors  on  one  side 
and  the  product  on  the  other,  are  valuable  aids  in  securing  readiness  and 
accuracy.  They  are  equally  usable  in  multiplication  and  factoring  (or 
division),  by  varying  the  sides  shown. 

In  learning  a  table,  it  should  be  kept  in  mind  that  the 
product  is  the  same  regardless  of  the  order  of  the  factors. 
Thus,  136  is  the  product  of  both  8x17  and  17  x  8.  If  this 
is  kept  in  mind,  the  new  combinations  to  be  learned  in  the 
higher  tables  become  constantly  less ;  e.g.  if  all  the  tables 
through  the  19's  are  known,  the  only  new  combination  in  the 
20's  to  be  learned  is,  20  x  20  =  400. 

After  completing  the  12's  to  20,  it  will  be  found  helpful 
to  pursue  the  following  order  : 

Review  the  5's  and  lO's  and  study  the  15's  and  20's. 
Review  the  3's,  6's,  9's,  12's,  and  take  the  18's. 
Review  the  4's,  8's,  12's,  and  take  the  16's. 
Then  take  the  13's,  14's,  17's,  and  19's. 

Problems  for  practice  in  short  multiplication  and  division 
with  abstract  numbers  may  be  drawn  from  the  above  table 
or  dictated  to  students  at  will. 

13.    Suggestions  and  Problems  for  Explanation. 

Problem.  — A  horse  worth  |130  and  3  cows  worth  $36 
each,  were  exchanged  for  sheep  at  $  6  per  head  and  $  82  in 
money.     How  many  sheep  were  received  ? 

136  X  3  =  1108. 
1108  +  f  130  =  $238,  value  of  horse  and  cows. 
1238 -$82  =  1156. 
$  156  -^  1 6  =  26,  the  number  of  sheep  received. 

Explanation.  —  If  1  cow  was  worth  1 36,  3  cows  were 
worth  three  times  $36,  or  $108.     If   the  cows  were  worth 
c 


18  ELEMENTS  OF  BUSINESS  ARITHMETIC 

$108  and  the  horse  1130,  together,  they  would  be  worth 
the  sum,  or  $238.  If  $82  was  received  in  money,  the  differ- 
ence between  $238  and  $82,  or  $156,  was  received  in  sheep. 
If  each  sheep  was  worth  $  6,  as  many  would  be  received  as 
$6  is  contained  in  $156,  or  26.     (See  Sec.  10.) 

Two  kinds  of  division  problems.  Division,  as  commonly 
used,  includes  two  classes  of  problems,  e.g. : 

(1)  When  it  is  required  to  find  how  many  times  one  quan- 
tity is  contained  in  another,  and 

(2)  When  it  is  desired  to  find  the  size  of  one  of  the  equal 
parts  of  the  quantity.  As  examples,  the  following  may  be 
given  : 

1.  How  many  bags  will  be  required  to  hold  42  bushels  of 
wheat,  if  each  bag  will  hold  3  bushels  ? 

2.  How  many  bushels  in  each  bag,  if  42  bushels  are  put 
into  14  bags  ? 

In  (1)  the  dividend  and  divisor  are  the  same  in  kind. 

In  (2)  the  dividend  and  the  result  are  the  same  in  kind. 

For  the  purposes  of  clear  thinking  and  rigid  explanation, 
it  is  thought  best  to  treat  problems  like  (2)  as  problems  in 
fractions. 

To  assist  in  keeping  the  reasoning  clear,  it  is  suggested 
that  problems  like  (1)  be  written  in  equation  form  in  this 
way : 

42  bu.  -r-  3  bu.  =  14,  the  number  of  bags  required. 

Problems  like  (2)  should  be  written : 

^^  of  42  bu.  =  3  bu.,  number  in  each  bag. 

It  should  be  noted,  however,  that  the  results  in  both  classes 
of  problems  are  obtained  by  the  same  process. 

In  the  written  solution  of  a  problem,  great  care  should  be 
taken  to  use  the  equation  correctly. 

An  equation  should  be  true  in  kind  as  well  as  in  amount. 


MULTIPLICATION  AND  DIVISION  19 

Concrete  numbers  should  invariably  be  named,  and  should 
be  written  first  in  the  equation. 

The  sign  of  multiplication  is  the  St.  Andrew's  Cross.  It 
should  be  read  "multiplied  by."  If  the  word  times  is  used, 
the  second  number  is  read  first.  Thus,  the  first  equation 
above  should  read,  %  36  multiplied  by  3,  or  3  times  1 36. 

Reading.  —  When  the  reasons  for  the  steps  of  the  prob- 
lem are  well  known  by  the  pupil,  much  time  may  be  gained 
by  a  mere  reading  of  the  equations  instead  of  requiring  an 
explanation. 

The  above  statement  would  be  read  as  follows :  $  36  mul- 
tiplied by  3  is  flOS;  1108  plus  1130  is  $238;  1238  less 
1 82  is  $  156  ;  and  1 156  divided  by  1 6  is  26,  or  the  number 
of  sheep  received. 

14.  Multiplication  by  Numbers  Greater  than  20.  Multi- 
plication, when  the  multiplier  is  greater  than  one's  known 
multiplication  table,  involves  the  use  of  partial  cnof^  >. 

products  or  the  product  of  each  figure  of  the  ^^^ 

multiplier  and  the  whole  of  the  multiplicand.        i.  rvj^r-r^ 
The   essential   thing   to   be    observed    in   the      -ic^oR 
process  is  that  the  right-hand  figure  of  each    Arj^c,^ 
partial  product  should  be  directly  under  each  '        ^  ^.-^  , 
multiplier.     The  reason  for  this  is  easily  seen 
when  we  remember  that  the  units  figure  is  multiplied  first 
in  each  case.     Thus,  units  times  units  give  units,  and  units 
multiplied  by  hundreds  are  hundreds,  etc.,  the  result  each 
time  being  of  the  same  denomination  as  that  of  the  figure 
multiplied  by.     The  fact  that  these  partial  products  are  to 
be  added  to  find  the  complete  product  explains  the  necessity 
of  arranging  units  under  units  and  tens  under  tens. 

When  the  multiplier  contains  a  cipher,  it  is  passed  over  and 
the  next  figure  to  the  left  becomes  the  multiplier.  Care  should 
be  taken  to  begin  the  partial  product  under  the  new  multiplier. 


20  ELEMENTS  OF  BUSINESS  ARITHMETIC 

When  ciphers  are  to  the  right  of  either  the  multiplier  or 
multiplicand,  or  both,  the  last  figures  to  the  right  of  both 
(other  than  the  ciphers)  are  placed  under  each  other.  The 
ciphers  are  disregarded  in  the  partial  products,  but  as  many 
are  placed  to  the  right  of  the  product  as  are  to  the  right  of 
both  numbers  being  multiplied. 

15.  Long  Division.  If  the  quotient  is  placed  above  the 
dividend,  the  reason  for  the  place  of  each  figure  will  be  more 
readily  seen.     Thus,  there  are  9  hundred  thirty-  ^rn 

fours  in  325  hundreds,  and  accordingly  the  9  is   04.^09^79 
placed  directly  above  the   hundreds  of  the  divi-         oa/> 
dend  ;   likewise  34  into  197  tens  gives  tens^  and         ~Tq7 
should   be  placed   directly  over   the  tens  of   the  -j,^^ 

dividend,  etc.  "979 

The  ease  with  which  one  gets,  to  know  the  de-  c)ncf 

nomination  of  any  partial  quotient  will  often  pre-  

vent  getting  absurd  answers,  and  will  greatly  simplify  the 
"  pointing  off  "  process  in  decimals  to  be  met  later. 

Problems  in  which  the  divisor  contains  ciphers  to  the 
right  may  be  much  shortened  by  cutting  off  as  many  figures 
to  the  right  of  the  dividend  as  there  are  ciphers  to  the  right 
of  the  divisor.  The  figures  so  cut  off  are  always  the  whole 
or  part  of  the  remainder,  and  if  written  fractionally  should 
be  written  over  the  entire  divisor. 


19^15. 


00 


15(5)9)  285)2^. 

The  process  of  long  division  may  be  ^  ,^^ 

much   shortened   by   subtracting    the  coyr^Aoo 

partial  products  as  we  find  them,  writ-  91  a 

ing   only   the    remainder   each    time.  090 

Drill  on  this  will  not  only  be  found  ^^^^ 

valuable  as  a  mental  training,  but  of  ^^            .    , 

.     ,          .             ,    ^  12,  remainder, 
real  value  m  lessening  work. 


MULTIPLICATION  AND  DIVISION  21 

PROBLEMS 

1.  7668  -  36.  11.  1,674,918  -  1980.  21.  372,104  -4-  386. 

2.  48,967  -  52.  12.  324,217  -  268.  22.  385,200  -^  1370. 

3.  20,982-78.  13.  356,686-682.  23.  466,830 -^ 2730. 

4.  24,476  -^  58.  14.  1,769,824  -  3800.  24.  8,326,900  -  1029. 

5.  10,908  -  236.  15.  367,240  -  461.  25.  9,230,021  ^  3120. 

6.  98,340-^63.  16.  378,625-325.  26.  780,164-119. 

7.  76,055-53.  17.  148,050-315.  27.  27,552 -f- 328. 

8.  42,400  -i-  98.  18.  986,172  -  186.  28.  172,096  -  3600. 

9.  39,628-76.  19.  34,572-129.  29.  34,216-^9203. 
10.  97,266  -f-  78.  20.  3,467,000  ^  360.  30.  1,647,756  -  198. 

Solve,  expressing  in  equation  form,  and  explain : 

1.  Bought  568  bushels  of  corn  at  48^  a  bushel  and  675  bushels  of 
wheat  at  76 J^  a  bushel.     What  did  both  cost? 

Note.  —  While  48^  is  the  real  multiplicand,  the  problem  is  shortened 
by  treating  it  (being  the  smaller  number)  as  the  multiplier.  In 
expressing  the  work  in  an  equation,  however,  48^  must  appear  as  the 
multiplicand. 

2.  Mr.  McFarland  has  property  valued  at  $6700.  He  buys  land 
and  sells  it  at  a  gain  of  1 5  per  acre.  He  is  then  rated  at  $8075.  How 
many  acres  did  he  buy? 

3.  A  stock  buyer  having  $3540  buys  16  horses  at  $75  each,  and 
invests  the  remainder  of  his  money  in  cattle  at  $36  per  head.  How 
many  head  of  cattle  does  he  buy  ? 

4.  At  the  rate  of  45  miles  an  hour,  how  long  will  it  take  a  train  to 
run  325  miles  ? 

5.  If  a  book  contains  255  pages,  and  there  are  1864  ems  to  the  page, 
how  many  ems  in  the  book  ? 

6.  An  electric  railway  company  has  435  miles  of  track  which  was 
built  at  a  cost  of  $  83,672  per  mile.  What  was  the  total  cost  of  con- 
structing the  road  ? 

7.  A  coal  dealer  bought  428  tons  of  coal  at  $  7.50  per  ton.  He  sold 
200  tons  of  it  at  $8  a  ton  and  the  remainder  at  $ 8.75.     Find  his  gain. 

8.  A  man  sold  mining  stock  for  $  9600,  a  mill  for  $  12,600,  and  three 
houses  for  $  12,530,  $  6780,  and  $  9870  respectively.     He  invested  t21-,^^r^===^ 


22  ELEMENTS  OF  BUSINESS  ARITHMETIC 

of  the  proceeds  in  the  stock  of  a  manufacturing  company  at  $75  a 
share,  and  the  balance  in  railroad  stock  at  $80  a  share.  How  many 
shares  of  each  did  he  buy? 

9.  If  a  clerk  receives  $1850  a  year,  and  pays  $38  a  month  for 
board  and  room,  $2.40  a  month  for  laundry,  $3.80  a  month  for  life  in- 
surance, $  12  a  month  to  a  building  and  loan  association,  $  376  a  year  for 
clothing,  and  $  325  a  year  for  incidentals,  how  much  does  he  have  to 
invest  in  business  after  a  term  of  ten  years  ? 

10.  Into  how  many  states  as  large  as  Texas  (265,780  sq.  mi.)  could 
the  United  States  (3,616,484  sq.  mi.)  be  divided? 

11.  A  manufacturer  bought  165  tons  of  steel  billets  at  $32  a  ton, 
45,000  pounds  of  steel  bars  at  $  1.80  a  hundred,  and  75  tons  pig  iron  at 
$  19  a  ton.     How  much  did  he  pay  for  all? 

12.  A  young  man  takes  out  a  life  insurance  policy  for  $2000  and 
agrees  to  pay  $3.78  a  month  for  20  years.  How  much  money  will  he 
have  paid  to  the  company  at  the  end  of  that  time  ? 

13.  How  many  cords  of  128  cu.  ft.  in  a  pile  of  wood  containing 
122,880  cu.  ft.  ?    AVhat  is  it  worth  at  $  3.50  per  cord  ? 

14.  A  dealer  bought  370  tons  of  coal  by  the  long  ton  (2240  lb.)  at 
$5  a  ton.  He  retailed  it  at  $7  a  short  ton  (2000  lb.).  What  was  his 
total  gain  ?  What  would  he  have  gained  if  he  had  sold  it  at  the  same 
price  per  long  ton? 

15.  A  stock  dealer  sold  26  carloads  of  cattle,  26  head  of  cattle  in  each 
car,  at  3;^^  a  lb.  If  the  cattle  averaged  935  lb.,  how  much  did  he  receive 
for  them? 

16.  I  paid  $  12,400  for  apples  at  $  2.50  per  barrel.  The  total  loss  in 
storage  was  38  barrels,  and  I  paid  5^  a  barrel  for  storage  and  2^  a 
barrel  for  drayage.  How  much  did  I  gain,  if  I  sold  them  for  $  3.80  per 
barrel  ? 

17.  A  certain  oil  company  produced  23,000,000  barrels  of  refined  oil 
in  one  year.  How  many  tanks  each  holding  35,000  barrels  would  it  take 
to  store  this  oil?  On  one  of  the  company's  farms  there  was  stored  in 
tanks  of  the  above  capacity  2,450,000  barrels  of  crude  oil.  How  many 
tanks  were  there  ? 

18.  If  the  area  of  all  the  continental  divisions  of  the  earth  is  51,238,800 
sq.  mi.  and  the  population  is  1,487,900,000,  how  many  people  are  there  to 
the  sq.  mi.  ? 


Ill 

DECIMALS 

16.  Decimal  Notation.  By  decimal^  is  meant  a  tenth.  Our 
notation  (Arabic)  is  said  to  be  a  decimal  notation  because 
each  denomination  is  one  tenth  of  the  next  larger  denomina- 
tion. Thus,  units  are  tenths  of  tens,  tens  are  tenths  of  hun- 
dreds, and  hundreds  are  tenths  of  thousands. 

Likewise  tenths  of  units  are  written  to  the  right  of  units, 
but  separated  from  them  by  a  decimal  point  (.).  Tenths  of 
tenths,  or  hundredths,  are  written  in  the  next  place  to  the 
right,  and  tenths  of  hundredths,  or  thousandths,  in  the  next 
or  third  place  from  the  decimal  point.  Thus,  324.516  con- 
sists of  3  hundreds,  2  tens,  4  units,  5  tenths^  1  hundredth^  and 
6  thousandths. 

While  the  system  is  thus  properly  termed  a  system  of 
decimal  notation,  the  meaning  of  the  word  decimal  is  made 
more  specific  by  restricting  its  use  to  the  part  of  a  number 
which  is  to  the  right  of  the  decimal  point ;  the  whole  num- 
bers to  the  left  being  termed  integers. 

It  will  therefore  be  seen  that  the  denominations  to  the 
right  of  units  are  the  same  as  those  to  the  left,  except  that 
they  decrease  by  tens  while  those  to  the  left  increase  by  tens. 
These  decreasing  denominations  are  distinguished  by  adding 
the  suffix  ths  to  each  denomination. 

17.  Reading  and  Writing  Decimals.  Beginning  at  the 
decimal  point,  decimals  are  read  exactly  as  whole  numbers  are 
read,  with  the  denomination  of  the  last  place  to  the  right 
named.     The  name  of  the  denomination  of  the  right-hand 


24  ELEMENTS  OF  BUSINESS  ARITHMETIC 

place  is  the  same  as  if  it  were  a  whole  number,  with  one 
additional  place,  and  with  the  suffix  ths  added.  The  word 
and  is  always  used  between  the  whole  number  and  the  deci- 
mal. Thus,  3,204.6103  is  read  :  three  thousand  two  hundred 
four  and  six  thousand  one  hundred  three  ten-thousandths. 

Write  decimals  exactly  as  whole  numbers  are  written,  and 
then  set  off  to  the  right  of  the  decimal  point  the  num- 
ber of  places  indicated  by  the  name  of  the  right-hand  de- 
nomination. The  number  of  places  to  be  set  off  is  always 
one  less  than  the  number  of  places  required  to  write  the  corre- 
sponding denomination  as  a  whole  number.  This  is  because 
the  units  place  is  always  to  the  left,  and  never  to  the  right, 
of  the  decimal  point.  Thus,  506  ten-thousandths  is  written 
w\t\i  four  places  (.0506)  because  ten  thousand  as  a  whole 
number  requires  five  places.  Ciphers  are  used  to  the  left  of 
the  number  to  be  written  as  a  decimal  when  necessary  to  set 
it  away  from  the  decimal  point.  In  whole  numbers  the  deci- 
mal point  is  presumed  to  be  at  the  right  of  units  figure, 
whether  it  is  written  or  not. 

Write  : 

1.  Sixty-eight  thousandths. 

2.  Five  tenths.     Seven  hundredths.     Eighteen  hundredths. 

3.  Six  thousandths.  Five  hundred  three  thousandths.  Eighteen 
thousandths.  Seven  hundred  thirty-five  ten-thousandths.  Eighty-four 
hundred-thousandths. 

4.  Eighteen  hundredths.  Thirteen  hundred-thousandths.  Sixty- 
four  tenths. 

5.  Eighty-five  ten-thousandths.  Three  hundred  twenty-three  and 
fifty-six  hundredths.  Seventy-eight  thousandths.  Three  hundred 
twenty-three  thousandths. 

Read: 

1.  .5,  .25,  .025,  .0625,  .0005. 

2.  .00001,  .3762536,  .00875,  .000025. 

3.  2.5,  25.25,  25.025,  700.07,  .303. 

4.  .193,  4.2,  6.3028,  .03275,  176.4. 


DECIMALS  25 

18.  Addition  and  Subtraction  of  Decimals.  The  processes 
and  principles  applied  in  the  addition  and  subtraction  of 
decimals  are  the  same  as  of  integers.  The  same  care  should 
be  exercised  in  keeping  like  denominations  in  the  same 
column.  The  decimal  point  in  the  sum  or  difference  is 
placed  directly  under  the  decimal  points  in  the  numbers 
added  or  subtracted. 

All  decimals  smaller  than  thousandths  are  usually  dropped, 
one  being  usually  added  to  the  thousandths  column  when 
the  discarded  decimal  is  five  tenths  or  more. 


PROBLEMS 

1.  7.17 

2.     6.789 

3. 

23.768 

4.     6.34 

5.     37.68 

.9 

.075 

.45 

.0074 

1.045 

.0006 

74.8 

6.985 

43.325 

174.632 

4.76 

.4 

17.005 

24.18 

35.986 

5.0017 

24.986 

3.178 

3.457 

1.706 

6.  .024  +  1.54  +  74.6  +  27.878  =  ? 

7.  234.96  +  756  +  40.5  +  6.03  +  1.005  =  ? 

8.  2.054  +  35.78  +  .067  +  .65  +  .268  =  ? 

9.  .9  +  13.564  +  234.96  +  8.5  +  .306  +  41.87  =  ? 

10.  .335  +  23.75  +  601.76  +  .007  +  35.86  =  ? 

11.  .91  +  13.564  +  234.96  +  8.5  +  .306  +  41.87  =  ? 

12.  l-.063  =  ?     13.435  -  .106  =  ? 

13.  3.5872  +  1.2834  =  ? 

14.  6-2.763  =  ? 

15.  8-3.234  =  ? 

16.  4.1  +  67.5  +  42.007  +  17.14  +  .0009  =  ? 

17.  34.006-15.556  =  ? 

18.  68.215-36.5  =  ? 

19.  94.35  -  36.7  =  ? 

20.  46.235  -  22.065  =  ? 


26  ELEMENTS  OF  BUSINESS  ARITHMETIC 

21.  216.745  -  176.89  =  ?  - 

22.  681.34  -  95.275  =  ? 

23.  14.367  +  743.65  +  .8  +  .306  +  9.845  +  834  +  7.63  =  ? 

24.  3.8  +  1.576  +  3.42  +  4  +  2.372  +  .8  +  354  =  ? 

25.  43.382  -  17.06785  =  ? 

19.    Multiplication  and  Division  by  1  with  Ciphers  Annexed. 

Since  the  value  of  each  place  in  both  whole  numbers  and 
decimals  is  one  tenth  of  that  to  the  left  of  it,  any  way  by 
which  each  figure  can  be  shifted  one  place  to  the  righU  or 
one  place  nearer  units,  will  decrease  its  value  to  one  tenth 
of  its  former  value.  Shifting  two  places  would  likewise 
decrease  it  to  one  hundredth,  three  places  to  one  thousandth, 
etc.  In  the  same  way  shifting  one  place  to  the  left  would 
multiply  it  by  ten,  two  places  by  one  hundred,  etc.  This 
may  be  accomplished  by  moving  the  decimal  point  to  the 
left  or  right.  Thus,  25  -j-  10  =  2.5  or  25  x  10  =  250,  and 
25  -^  1000  =  .025  or  .25  x  1000  =  250. 

In  multiplying  hy  1  with  ciphers  annexed^  the  decimal  point 
is  moved  to  the  right  as  many  places  as  there  are  ciphers  in  the 
multiplier.     Thus, 

326.02x100    =32602, 
and  100.2437  x  1000  =  100243.T. 

In  dividing  hy  1  with  ciphers  annexed^  the  decimal  place  is 
moved  as  many  places  to  the  left  as  there  are  ciphers  in  the 
divisor.     Thus, 

326.02-^100  =  3.2602, 
and  3.42 -^100  =  .0342. 

These  simple  facts  are  of  great  convenience  in  performing 
operations  with  decimals  and  in  shortening  multiplication 
and  division  of  whole  numbers. 

Note.  —  Problems  like  those  above  should  be  given  until  the  pupil 
thinks  of  no  other  way  of  multiplying  or  dividing  by  10, 100,  etc.,  than  by 


DECIMALS  27 

removing  the  decimal  point  the  proper  number  of  places  to  the  right  or 
left. 

20.   Multiplication  of  Decimals. 

(a)  9  times  .25  =  2.25. 

When  a  decimal  is  multiplied  by  a  whole  number,  the 
denomination  of  the  decimal  of  the  product  is  the  same  as 
that  of  the  multiplicand. 

(h)  .1  times  9  means  one  tenth  of  9  or  .9  (Sec.  19). 

.25  x.l=.025. 

Multiplying  by  .1  is  the  same  as  dividing  by  10;  by  .01 
the  same  as  dividing  by  100,  etc.  (Sec.  19). 

(c)  .25  X  .7  =  ?  (t^)  .25  X  6.25=  ? 

.25x.l  =  .025  .25 X    .01  =  .0025  (see 5) 

.025  X   7  =  .175  .0025x  625  =  1.5625  (see a) 

In  multiplying  decimals  by  tenths,  hundredths,  etc.,  one 
tenth,  one  hundredth,  etc.,  is  found,  and  the  result  is  then 
multiplied  by  the  number  of  tenths,  hundredths,  etc.,  in  the 
multiplier.     Or, 

Remove  the  decimal  place  in  the  multiplicand  as  many  places 
to  the  left  as  there  are  decimal  places  in  the  multiplier^  then 
multiply  by  the  multiplier  as  a  whole  number. 

PROBLEMS 

1.  214.76  X  89.104  6.   24.075  x  16 

2.  3.0046  X  43.25  7.  45.009  x  78 

3.  .8756  X  .173  8.  50.13  x  4.321 

4.  .045  X  18  9.  176.84  x  4.321 

5.  64  X  .032  10.  95.817  x  1000 

Find  the  cost  of : 

11.  24,800  bricks  @  $7.35  per  M. 

12.  875  lb.  hay  @  11.25  per  hundredweight. 

13.  186  bu.  wheat  @  67^^  per  bushel. 


28  ELEMENTS  OF  BUSINESS  ARITHMETIC 

14.  15,680  lemons  at  65^  per  hundredweight. 

15.  357  bu.  oats  @  30^  ^  per  bushel. 

16.  75  bbl.  flour  @  %  4.15  per  barrel. 

17.  70  bbl.  mess  pork  @  $  10.50  per  barrel. 

18.  14  bbl.  beef  @  $14.40  per  barrel. 

19.  5  cases  shredded  codfish  @  %  4.90  a  case. 

20.  3  cases  canned  pineapples,  6  doz.  each,  @  %  2.87|  per  dozen. 

21.  30  bbl.  mess  beef  @  $  14.85  per  barrel. 

21.  Division  when  the  Divisor  is  an  Integer. 

Solve  : 

1.  .6  -4-  2  =  ?    Just  as  6  bu.  -^  2  =  3  bu.  (|  of  6  bu.),  so 
6  tenths  -2  =  3  tenths  or  .6  h-  2  =  .3. 


2.   2.6-^-2=? 

2.6-2  =  1.3. 

2)2.6 
1.3 

3.   .396-3 

5.   6.6  -  4 

7. 

.7236  -  9 

9. 

.16-5 

4.   12.15-3 

6.   3.68-4 

8. 

.084  -  4 

10. 

.54-6 

In  division  the  decimal  point  is  written  in  the  quotient  when 
it  is  reached  in  the  dividend.  In  short  division  it  should  be 
placed  directly  below  the  decimal  point  of  the  dividend ;  in 
long  division  directly  above. 

Illustrations. 

.06  1.24  .072 

{A)  5).18       (5)  15) .09       (C)  125)7.50     {D)  26)32.24     {E)  24)1.728 

.036  .006  7.50  62  48 

104 

From  inspection  of  problems  Ay  B^  and  j^,  just  above,  it 
will  be  seen  that  each  figure  in  the  dividend^  to  the  right  of 
the  decimal  pointy  requires  a  figure  (^or  cipher^  in  the  quotient. 

Ciphers  may  be  added  at  will  to  the  right  of  a  decimal 
without  altering  the  value.  They  are,  therefore,  annexed 
where  necessary  to  permit  further  division,  but  in  practice 
they  are  carried  in  the  mind  only  and  not  written. 


DECIMALS  29 

In  problems  D  and  E^  the  partial  products  are  omitted, 
the  subtraction  being  performed  mentally  (Sec.  15). 


So 
1. 

ilve: 
48.24  -  3 

10. 

1.5-4 

19. 

174.9  -4-  75 

2. 

8.64^4 

11. 

.06  -  15 

20. 

43.58  -4-  671 

3. 

.465  -^  3 

12. 

8.06  -  5 

21. 

345.9  -4-  329 

4. 

8.4-4-5 

13. 

.2^8 

22. 

56.89  -  137 

5. 

.648  -  6 

14. 

34.75  -4-  25 

23. 

.0789  -f-  703 

6. 

.114  -  7 

15. 

.0543  -4-  15 

24. 

.6789  --  212 

7. 

.81-^9 

16. 

.0255  -4- 11 

25. 

17.68  ^  245 

8. 

.63-9 

17. 

3.184-4-482 

26. 

334.4^76 

9. 

.12-4-9 

18. 

37.86  -  541 

22.    When  the  Divisor  contains  a  Decimal. 

Example:  1.728-^.12  =  ? 

If  we  could  change  the  divisor  .12  to  a  whole  number  and 
solve  as  in  Sec.  21,  the  problem  of  the  location  of  the 
decimal  point  in  the  quotient  (the  only  way  in  which  divi- 
sion of  decimals  differs  from  division  of  whole  numbers), 
would  be  an  easy  one. 

But  dividing  by  one  tenth  is  equivalent  to  multiplying  by 
ten,  dividing  by  one  hundredth  to  multiplying  by  one  hun- 
dred, etc.  (Sec.  20).  Thus  dividing  1.728  by  .01  =  1.728  x  100 
or  172.8  (Sec.  19),  and  dividing  by  .12  would  give  -^  of 
the  result  obtained  through  dividing  by  .01.     Thus : 

1.728 -f-. 01  =  172.8, 
and  172.8-1-  12  =  14.4. 

In  division  by  a  decimal,  therefore,  the  decimal  point  of  the 
divisor  should  he  moved  to  the  right  of  the  last  digits  and  that  of 
the  dividend  an  equal  number  of  places  to  the  right  of  its  origi- 
nal position  (Sec.  19).  Division  should  then  be  performed 
as  in  Sec.  21,  marking  the  decimal  point  in  the  quotient 
when  the  decimal  point  is  reached  in  the  dividend. 


30  ELEMENTS  OF  BUSINESS  ARITHMETIC 

To  avoid  losing  the  identity  of  the  original  divisor  and  to 
be  able  to  fix  the  exact  remainder,  should  there  be  one,  the 
position  of  the  new  decimal  point  should  be  indicated  only. 
A  small  St.  Andrew's  cross  (  X )  forms  a  convenient  mark 
for  that  purpose. 


Solve : 

1.   6.336^1.44. 

6. 

784  -^  4.235. 

17. 

1885  --  28.47. 

7. 

.3416  -f-  .0189. 

18. 

363.71  -^  1.126. 

4.4 

8. 

12.347  -f-  .0074. 

19. 

83.078  -^  3.57. 

1.44x)6.33x6 

9. 

.8765  -^  3.422. 

20. 

137.854  --  .425. 

5  76 

10. 

9  --  102. 

21. 

38.9007  --  .425. 

57  6 
57  6 

11. 

14.3768  -r-  .9817. 

22. 

568.148  -r-  201.03. 

12. 

84.45  -  .089. 

23. 

.81769  --  .0008175. 

2.   7.345 -.29. 

13. 

3894.78  -i-  4287. 

24. 

$135  ^$.37^. 

3.   250.754-^6.17. 

14. 

346.543  -  634.08. 

25. 

74  --  .0136. 

4.   6.0534-^-19.23. 

15. 

34.25  -^  84.6. 

26. 

.33614  -f-  13.45. 

5.   132.5  -=-  734. 

16. 

9.1342  -f-  208.4. 

27. 

18.3467  --  1.233. 

PROBLEMS 

1.  Add  eight  and  nine  tenths ;  seven  hundred  twenty-six  and  twenty- 
five  hundredths  ;  one  hundred  sixty-eight  and  ninety-seven  hundredths  ; 
one  thousand  three  and  seven  tenths;  seven  hundred  sixty-eight  and 
seventy-four  hundredths. 

2.  967.45  -f  8.674  -f  23.997  +  864.325  +  37286  -f  42.1  +  6.5  =  ? 

3.  From  six  thousand  seven  take  one  thousand  two  hundred  twenty- 
eight  and  seven  hundredths. 

4.  What  is  the  sum  of  3.25,  1.8,  67.89,  .0032,  879.435,  and  23.067? 

5.  What  is  the  sum  of  .125,  1.25,  12.5,  125,  .0125? 

6.  From  675.  take  67.893.  9.   Multiply  543.002  by  18.6. 

7.  From  345.6703  take  43.52.  10.   Divide  6423.38  by  28.87. 

8.  Multiply  54.054  by  .0678.  11.   Divide  .00684  by  .25. 

12.  At  $  9.25  per  ton,  how  much  coal  can  be  bought  for  $  67.53  ? 

13.  If  a  barrel  of   apples  cost  $5.15,  how  many  can  be  bought  for 
$258.75? 


DECIMALS  31 

14.  At  $  .26  per  dozen,  how  many  eggs  can  be  bought  for  $  185.32  ? 

15.  At  1 6.45  per  ton,  how  many  tons  of  soft  coal  can  be  bought  for 
$175? 

16.  A  farmer  sold  65  bu.  wheat  at  $.62|,  34  bu.  rye  at  $  .58^,  78  bbl. 
of  apples  at  $  6.40.  He  bought  35  lb.  sugar  at  $  .06,  25  gal.  molasses  at 
$  .85,  and  a  set  of  harness  at  $  16.75.     How  much  money  had  he  left  ? 

17.  A  carpenter  earned  $  15.60  a  week  for  8  weeks.  The  first  week 
he  spent  $8.75,  the  second  week  $11.45,  and  the  remaining  weeks  he 
spent  $  8.50  per  week  on  an  average.  How  much  money  had  he  at  the 
end  of  the  time  ? 

18.  A  man  who  has  an  income  of  $  6785  per  year  spends  $  1385.75. 
How  much  does  he  save? 

19.  A  hardware  merchant  had,  at  the  beginning  of  the  year,  $  9800. 
During  the  year  he  bought  goods  to  the  amount  of  $  7845,  and  sold  to 
the  amount  of  $  7856.  If  the  goods  he  had  on  hand  at  the  end  of  the 
year  were  worth  $  8340.65,  what  did  he  gain  or  lose  during  the  year  ? 

20.  A  man  bought  a  mower  for  $  67,  a  wagon  for  $  56.50,  a  plow  for 
$6,  and  a  rake  for  $1.75.  If  he  gave  the  merchant  two  one-hundred- 
dollar  bills,  how  much  change  did  he  receive  ? 

21.  Mr.  A.  W.  Springer  bought  of  C.  E.  Dunlap,  75  bbl.  flour  at  $5.14, 
and  34  bbl.  buckwheat  flour  at  $3.95.  What  was  the  amount  of  the 
biU? 

22.  Mr.  S.  H.  Detmore  bought  of  Wm.  King,  124  boxes  oranges  at 
$  1.12|,  12  boxes  figs  at  $  1.62^,  33  boxes  apricots  at  $  2.15,  15  boxes 
citrons  at  $  1.33|,  and  12  boxes  layer  raisins  at  $2.95.  Find  the  amount 
of  the  bill. 

23.  How  many  cases  tomatoes,  70  doz.  to  the  case,  at  92  ^  per  dozen, 
can  be  bought  for  $  277.55  ? 


IV 
FRACTIONAL  PARTS 

23.  Use  of  Fractional  Parts.  The  necessities  of  everyday 
business  require  the  use  of  few  processes  in  arithmetic  more 
frequently  than  that  of  finding  some  part  of  a  given  quantity. 
Not  only  is  this  true  in  the  simpler  problems  of  business,  but 
the  processes  of  percentage,  discount,  partnership,  interest, 
taxes,  etc.,  are,  fundamentally,  finding  fractional  parts. 

Finding  the  fractional  parts  requires  the  use  of  the  proc- 
esses of  multiplication  and  division,  together  with  the  simpler 
principles  of  decimals.  Under  the  usual  arrangement,  it 
forms  a  case  in  fractions ;  viz.,  multiplication  by  a  fraction. 
Its  common  use  and  wide  application  is  thereby  obscured, 
and  the  simplicity  of  the  operation  seldom  realized.  Then, 
too,  such  problems  are  commonly  included  in  division  and 
cause  a  great  deal  of  haziness  in  reasoning.  It  is  believed 
that  the  claims  of  clear  thinking,  and  the  importance  of  the 
power  sought,  warrant  treatment  in  a  separate  chapter. 

24.  Finding  a  Single  Fractional  Part. 
Find  J^  of  291.9.     (See  Sec.  13.) 

20.85    Ana. 
14)291.9 

28 

119 
112 


70 

70 
Find: 

1.  ^ff  of  5763  (Sec.  19),  320,  2000,  17,722,  1023,  1879,  98,450. 

2.  ^  of  150,  337,  20.0400,  5061,  17,002,  279.63,  56,840,  96.789. 

32 


FRACTIONAL  PARTS  33 

3.  i^  of  740  (i  of  yij),  648,  10.25,  2047,  50.011,  3452,  6.073,  3004. 

4.  i  of  450,  645,  705,  64.2,  687,  1018,  50,670,  58,605, 1600.115. 

5.  ^  of  7218,  6345,  117,927,  14.4639,  5017,  72668.54,  576,943. 

6.  I  of  588,  1672,  976,  847,  734,68^p^r33,  20,053,  1,129,734. 

7.  \  of  3876,  327,450,  389.90,  263,87^2^,  32,876,  42,680. 

8.  \  of  42,744,  62,728,  88,102.  \  of  67,833,  300.0321,  67431.76318. 
^^  of  27,924,12,  102,603.  ^V  of  367,485,  790,532.  J^  of  362,922,  758.492. 
yig  of  786,496,  710.43,  876,014.     f^  of  72,632,  43.2004,  432,132. 

9.  ^  of  7364,  5467,  18,998,  and  6555.88.  ij  of  4763,  9878,  190.63,  and 
121,264.396.     ^V  of  28,652.  10,431,278  and  624.0156. 

25.  Divisibility.  The  following  facts  will  be  found  help- 
ful in  finding  fractional  parts  and  in  short  division. 

1.  When  the  right-hand  digit  of  a  number  is  divisible  by 
2,  the  whole  number  is  divisible  by  2.  When  exactly 
divisible  by  2,  a  number  is  said  to  be  even;  when  not  so 
divisible,  it  is  odd. 

2.  When  the  right-hand  digit  of  a  number  is  5  or  0,  the 
whole  number  is  divisible  by  5;  when  it  is  0,  the  whole 
number  is  divisible  by  10. 

3.  When  the  number  expressed  by  the  two  right-hand 
digits  is  divisible  by  4,  the  whole  number  is  divisible  by  4. 
When  the  number  expressed  by  the  three  right-hand  digits 
is  divisible  by  8,  the  whole  number  is  divisible  by  8. 

4.  When  the  sum  of  all  its  digits  is  divisible  by  3  or  9, 
the  number  is  divisible  by  3  or  9. 

5.  When  an  even  number  is  divisible  by  3,  it  is  also 
divisible  by  6. 

26.  Factors  and  Multiples.  A  number  exactly  divisible  by 
another  number,  other  than  1,  is  said  to  be  composite^  ^.e., 
composed  of  other  numbers  called  factors^  which  multiplied 
together  will  produce  the  number. 

A  number  which  is  not  exactly  divisible  by  another  (ex- 
cept by  unity),  is  said  to  be  prime. 


34  ELEMENTS  OF  BUSINESS  ARITHMETIC 

Factors  may,  themselves,  be  either  prime  or  composite. 

A  number  which  will  exactly  divide  two  or  more  numbers 
is  said  to  be  a  common  divisor  or  a  com,mon  factor.  The 
largest  number  which  will  exactly  divide  two  or  more  num- 
bers is  their  greatest  common  factor  or  greatest  common  divisor. 
If  two  or  more  numbers  have  no  common  divisor,  they  are 
said  to  be  relatively  prime. 

A  number  which  is  two  or  more  times  another  number  is 
its  multiple.  When  a  number  is  a  multiple  of  two  or  more 
numbers,  it  is  their  common  multiple.  When  it  is  the  smallest 
number  which  is  a  multiple  of  two  or  more  numbers,  it  is 
their  least  common  multiple. 

27.  Averaging.  By  average  is  meant  the  size  of  a  part  if 
the  sum  of  a  given  series  of  numbers  is  distributed  into 
equal  parts.  Thus,  we  say  the  average  cost,  average  monthly 
expenses,  average  daily  attendance,  the  assessed  valuation 
per  capita.,  etc. 

In  business,  we  may  more  clearly  judge  given  conditions  by 
the  use  of  an  average  cost,  output,  etc.,  than  if  we  were  com- 
pelled to  keep  in  mind  the  several  amounts.  In  administra- 
tive statistics  a  more  intelligible  summary  of  facts  may  often 
be  made  by  using  averages.  In  many  similar  ways  averaging 
enters  into  business,  and  it  should  be  early  understood. 

Problem.  —  A  merchant's  receipts  for  one  week   were  : 
$140.45,   1217.20,   $200,  $209.80,    $432.75,   and  $630.40. 
Find  his  average  daily  receipts. 
Solution.  —$140.45 
217.20 
200.00 
209.80 
432.75 
630.40 
\  of  $1830.60  =  $305.10,  average  daily  receipts. 


FRA.CTIONAL  PARTS 


35 


The  general  method  of  averaging  seen  in  the  above  is  to 
find  the  sum  of  the  quantities  to  be  averaged,  and  divide 
that  sum  into  as  many  equal  parts  as  there  are  quantities 
added. 

PROBLEMS 

1.  If  a  man's  gains  for  the  year  were  as  follows:  $628,  $75,  $220, 
$865,  $2205,  $3600,  $1780,  $1500,  $1240,  $3275,  $825,  $775,  what  was 
his  average  monthly  gain? 

2.  A  grain  dealer  bought  during  the  week:  2600,  3850,  4506,  7870, 
9675,  and  5490  bushels  of  wheat.  What  was  the  average  daily  purchase? 
He  paid  the  market  prices  each  day,  which  were  quoted  at  85^,  85^  )2^, 
88)25,  90)*,  S4:^,  and  80 (2^.  What  was  the  average  cost  per  bushel?  How 
would  he  know  what  price  per  bushel  to  ask  that  he  may  not  lose  in  the 
transaction  ? 

3.  An  agent's  expenses  were  as  follows :  January,  $  125,  February, 
$75,  March,  $80,  April,  $95,  May,  $105,  and  June,  $225.  What  were 
his  average  monthly  expenses? 

4.  Find  the  total  time,  the  amount  due,  and  the  average  daily  wages 
for  each  laborer  in  the  following  time  sheet,  counting  8  hrs.  to  the  day ; 
also  find  the  average  wage  for  all. 

Time  Sheet  for  Week  Ending  April  30, 19—. 


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5.   A  student's  grades  for  his  high   school  course  were  as  follows: 
Algebra  92,  Geometry  86,  Beginning  Latin  89,  Caesar  94,  Cicero  94, 


36  ELEMENTS  OF  BUSINESS  ARITHMETIC 

Virgil  92,  German  90,  Literature  83,  Essays  85,  Chemistry  92,  Physics 
90,  Physiology  95,  Greek  History  87,  Roman  History  88,  U.  S.  History 
87,  Arithmetic  85,  Bookkeeping  90.  What  was  his  average  grade  for  the 
course  ? 

28.  To  find  the  Value  of  more  than  One  Part.  Tenths. 
(Sec.  19.)  To  find  -^-^  take  \.  To  find  any  other  number  of 
tenths,  find  J^  as  in  section  19,  and  multiply  by  the  number 
of  tenths  desired. 

Solve : 

i\  of  323,496,  T-^  of  769,284,  ^  of  6789.40,  xV  of  81,145,  ^  of  72,910,  ^ 
of  432,561,  i-^  of  .63245,  t%  of  237,684. 

Twentieths.  (|-  of  1  tenth.)  J^  =  i,  -f^  =  -J^,  etc.  The 
simplest  form  of  the  fraction  should  be  taken  and  that  frac- 
tion of  the  number  then  found.  Practice  until  student  uses 
the  simplest  form  at  first  sight. 

Solve : 

1.  3j%of360  5.  ^^  of  3240 

2.  la  of  8.460  6.  if  of  7.380 

3.  5^  of  3750  7.  1^  of  960 

4.  ^5^  of  7260  8.  f|of490 

Fifths.  |  =  A,|  =  TVand-|  =  ^V 
Fourths,  f  =  J.  For  |,  subtract  \. 
Sixths      -^  =  -1    ^  =  1    4  _  2 

toiJ^LUS.        62'   63'   63* 

Eighths,     f  =  i   2  ^  i^  |  =  |^  etc. 

Twelfths,     -f^  =  ^>  iV  =  h  -\2  —  h  ®^^*     ■^^^'  11  subtract  ^2 • 
In   general,    use   simpler    equivalent    fractions   wherever 

possible.     If  there  are  none,  find  the  value  of  one  part  and 

multiply  by  the  number  of  parts  desired. 

Solve : 

1.  T^Yof726  4.  I  of  911  7.  t'^j  of  728 

2.  ^^  of  543  5.  ^5^  of  614  8.  f  of  329 

3.  I  of  789  6.  i3T0f729  9.  f  of  628 


9. 

H  of  4260 

10. 

^  of  428 

11. 

M  of  5.061 

12. 

A  of  576 

FRACTIONAL  PARTS  37 


10. 

j\  of  7028 

17. 

j\  of  48,689 

24. 

1  of  36,428 

11. 

j\  of  7028 

18. 

iV  of  45,186 

25. 

T%  of  7829 

12. 

jS  of  328.65 

19. 

j\  of  .3876 

26. 

r%  of  1.7206 

13. 

35V  of  16,924 

20. 

j%  of  4263 

27. 

f  of  72,963 

14. 

j%  of  6724 

21. 

1  of  67.908 

28. 

j\  of  34,102 

15. 

j%  of  72,365 

22. 

f  of  7252 

29. 

j\  of  4291 

16. 

j\  of  98,642 

23. 

f  of  81.720 

30. 

j\  of  28,367 

Solve : 

1. 

a-  of  68,734 

11. 

j\\  of  19,643 

21. 

j\\  of  47.6297 

2. 

If  of  57,836 

12. 

U  of  42,782 

22. 

fW  of  62,587 

3. 

H  of  196,732 

13. 

If  of  32,785 

23. 

/^«j  of  29,410 

4. 

j%\  of  67.941 

14. 

H  of  42,363 

24. 

7^^  of  25,000 

5. 

If  of  94,362 

15. 

^  of  89.642 

25. 

if'j  of  67,894 

6. 

II  of  6946 

16. 

^Vi7  of  2632 

26. 

iH  of  .456781 

7. 

j^^j  of  198,674 

17. 

if  of  15,922 

27. 

n  of  86,543 

8. 

H  of  .4575 

18. 

rV5  of  58,596 

28. 

f  1  of  219,876 

9. 

tV^  of  9.7176 

19. 

tV^  of  155,476 

29. 

^  of  98,634 

10. 

/A  of  32,642 

20. 

T«^  of  167.1918 

30. 

xVx  of  186,792 

29.  Simplified  Processes.  It  often  happens  that  the  num- 
bers to  be  multiplied  or  divided  are  such  that  the  operations 
of  multiplication  and  division  may  be  much  shortened.  The 
simplicity  of  the  shortened  method,  too,  often  lends  itself  to 
clearer  thinking  and  lessens  the  liability  of  error.  The  fol- 
lowing suggestions  are  not  given  to  be  learned  as  "short 
methods,"  but  as  examples  of  what  may  be  done  to  simplify 
problems  if  a  little  thinking  is  done  before  the  pupil  plunges 
into  the  mechanics  of  a  solution.  They  should  form  a  part 
of  the  methods  habitually  employed  and  not  merely  referred 
to  when  longer  processes  have  been  drilled  into  habit  by 
much  practice.  A  keen  appreciation  of  the  possibilities  for 
shortening  and  simplifying  many  of  the  arithmetical  pro- 
cesses will  lead  to  quick  work  and  accurate  results. 


38  ELEMENTS  OF  BUSINESS  ARITHMETIC 

Illustrations  : 

1.  426x100=?     By  1000  =  ?     By  10  =  ?     (Sec.  19.) 

Suggestion.  In  multiplying  by  a  number  consisting  of  1  with 
ciphers  annexed,  the  product  is  found  to  consist  of  the  multiplicand  with 
the  ciphers  annexed.     Thus,  728  x  1000  =  728,000. 

2.  53,648  ^  1000  =  ?     (Sec.  19.) 

Since  the  divisor  will  take  out  even  thousands,  the  hun- 
dreds, tens,  and  units  figures  will  form  the  remainder.  Thus, 
53,648^1000=53  and  -j^*^,  or  53.648.  Likewise,  8269 
-i- 100  =  82  and  {-^%  or  82.69. 

3.  7268  X  25  =  ? 

Multiplying  by  25  would  give  ^  as  much  as  multiply- 
ing by  100.  Thus,  the  above  may  be  written  at  sight  as 
182,300  (1  of  726,800).  Likewise  7268x50  =  1  of  726,800 
or  363,400. 

4.  742-25  =  ? 

The  quotient  is  manifestly  4  times  what  it  would  be  if 
100  were  the  divisor.  In  practice,  multiply  first  and  divide 
by  100.  Thus,  742  x  4  =  2968,  which  divided  by  100  =  29 
and  ^%8_  or  29.68.     Likewise  742  ^  50  =  7.42  x  2  or  14.84. 

5.  246  X  121  =  ?     (1  of  24,600.) 

6.  368^121=?     (368x8-^100.) 

7.  1284x15  =  ? 

If  the  table  of  15's  has  not  been  learned,  the  usual  process 
may  be  shortened  by  annexing  a  cipher  and  adding  J  of  the 
number  thus  formed  to  itself.     Thus, 

12,840 

6420 

19,260 

8.    Divide  5286  by  20.     By  30. 
The  quotient,  dividing  by  20,  is  clearly   -J   as   much   as 
when  divided  by  10.     Thus,  J  of  528.6  =  264.3. 


FEACTIONAL  PARTS  39 

9.    8246  X  98  =  ? 
It  is  evident  that  the  product  would  be  100  times  the  mul- 
tiplicand less  twice  the  latter.     Thus, 


824,600 

16,492 

808,108 

To 

multiply  by 

11 

or  any  multiple 

of  11 

328  X 

11 

=  ?     476x66  = 

:? 

10. 


To  multiply  by  11  is  to  multiply  by  10  +  1.  Therefore, 
multiply  328  by  10  and  add  328  ;  or  add  the  digits  as  fol- 
lows :  8;  8  +  2  =  10;  2  +  3  +  1  (carried)  =  6 ;  bring  down  3. 
The  result  is  3608. 

66  is  11  times  6.  Multiply  476  by  11,  and  that  by  6.  It 
may  be  done  as  follows  :  6  x  6  =  36  ;    write  6  and  carry  3. 

6  +  7  =  13;  6x13  +  3  (carried)  =  81 ;  write  1  and  carry  8. 

7  +  4  =  11;  6x11  +  8  (carried)  =  74  ;  write  4  and  carry  7. 
6x4  +  7  (carried)  =  31 ;  write  31.     The  result  is  31,416. 

11.  When  one  part  of  the  multiplier  is  contained  in  an- 
other, the  multiplication  may  be  shortened  as  shown  in  the 
following ; 

342  X  248  =  ?     167  x  412  =  ? 

Since  24  is  3  times  8,  it  is  evident  that  if  342  be  multi- 
plied by  8  and  that  result  by  3,  the  final  result  will  be  the 
same  as  though  we  multiplied  342  by  248  in  the  usual  man- 
ner (a). 

167  multiplied  by  4  gives  668.     668  mul- 
tiplied by   3    (12-^4)   gives   2004.      The        

final  result  is  the  sum  of  668  written  in       2,736    668 
hundreds'  place  and  2004  in  units'  place,  or     82,08         2004 
68,804  (5).  84,816    68,804 

12.  52x225  =  ? 


(a) 

(^) 

342 

167 

248 

412 

40  ELEMENTS  OF  BUSINESS  ARITHMETIC 

In  the  multiplier  225,  we  see  twice  and  ^  of  100  times  52, 
or  10,400 

1,300 
11,700 

13.  7558x125  =  ?^   |  of  (7558  x  1000). 

14.  57,632 -f- 370  =  5763.2 -V- 37. 

Solve : 

1.  37,685  X  10  11.   32,764  x  12^  21.  16,324  x  95 

2.  49,652  X  100  12.   25,670  x  742  22.  17,264  x  320 

3.  29,627  X  1000  13.   62,482  x  250  23.  98,643  -  20 

4.  72,864x25  14.   41,628x75  24.  76,523x300 

5.  22,345  X  15  15.   29,863  x  99  25.  8629  h-  12i 

6.  28,364  X  125  16.   36,486  x  175  26.  37,520  -  25 

7.  48,627  X  55  17.   28,634  x  325  27.  4286  x  88 

8.  26,327  X  98  18.   67,450  x  217  28.  36,742  x  97 

9.  4652  X  225  19.   6284  x  1500  29.  75,267  x  420 
10.  63,245x30  20.  729x40  30.  2672-50 

30.   Decimal  Equivalents  of  the  More  Common  Fractional 
Parts. 

Solve : 

1.  ^  =  how  many  tenths  ? 

i  =  ^  of  10  tenths  =  2  tenths  or  .2. 

2.  J  =  how  many  hundredths? 

^  =  i  of  100  hundredths  or  .25. 

3.  i  =  how  many  thousandths  ? 

^  =  i  of  1000  thousandths  or  .125. 

4.  ^  =  how  many  tenths?     How  many  hundredths? 

5.  ^  =  how  many  tenths  ?     How  many  hundredths  ? 

6.  ^  =  how  many  tenths?     How   many  hundredths?     How  many 
thousandths  ? 

^  =  ^  of  ten  tenths  or  .3|. 

7.  J  =  how  many  tenths?      How  many  hundredths?     How  many 
thousandths  ? 


FRACTIONAL  PARTS  41 

8.  ^  =  how  many  tenths?     How   many   hundredths?     How   many 
thousandths  ? 

9.  ^  =:  how  many  tenths?     How  many  hundredths?     How   many 
thousandths  ? 

10.  j*^  =  how  many  tenths?  How  many  hundredths?  How  many 
thousandths? 

11.  2V  =  I'ow  many  tenths?  How  many  hundredths?  How  many 
thousandths  ? 

12.  ^i^  =  how  many  tenths?  How  many  hundredths?  How  many 
thousandths  ? 

Table  of  decimal  equivalents.     Memorize. 
\  =  .2  i  =  .125  |  =  .14f 

Name  Decimal  Equivalents  at  Sight  : 

1.  I  5.   }  9.   x% 

2.  f  6.   I  10.   I 

3.  f  7.    f  11.    11 

4.  f  8.    I  12.    f 

31.  Fractional  Parts  of  One  Dollar. 

The  more  common  fractional  parts  of  $1  are : 

50^= fi      io^=iJ^     ^^^=^      61^=1^5 

The  more  usable  multiples  of  these  parts  are : 

371^  =  $!         871^  =  ||-  75^  =  ||  60^  =  If 

621^  =  If  66|^=lf  40^  =  $1  80^  =  || 

32.  Finding  the  Cost  of  Articles.  A  large  part  of  the 
multiplication  in  business  consists  in  finding  the  cost  of 
articles  at  a  given  price  each  or  per  dozen,  As  prices  are 
in  a  large  number  of  cases  a  simple  fractional  part  of  one 


^  = 

•08J 

■h  = 

.05 

^  = 

.04 

13. 

f 

17.   1 

14. 

f 

18.   f 

15. 

^ 

19-   iV 

16. 

f 

20.    1 

42  ELEMENTS  OF  BUSINESS  ARITHMETIC 

dollar,  this  fact  may  be  used  to  shorten  the  work,  and  what 
is  of  greater  importance,  to  secure  greater  accuracy  in  result. 
Thus, 

(a)  Find  cost  of  344  yd.  of  calico  at  12|^^  per  yard.  At 
12  J  ^  it  would  cost  I  as  much  as  it  would  at  81.  I  of  $344 
=  143. 

Note.  —  In  problems  like  this  where  the  multiplicand  is  an  easier 
number  to  multiply  by  than  the  real  multiplier,  it  is  so  used,  keeping  in 
mind  the  denomination  of  the  product. 

(h)  Paid  1403  for  corn  at  20^  per  bushel.  How  many 
bushels  did  I  buy  ? 

403  bu.  X  5  =  2015  bu. 

The  number  of  bushels  bought  is  clearly  5  times  greater 
than  if  it  were  worth  $  1  per  bushel. 

Find  Cost  of: 

1.  7286  bu.  wheat  @  50)*. 

2.  1456  yd.  prints  @  12^  ^. 

3.  764  yd.  cloth  @  33^^. 

4.  324  bbl.  mess  beef  @  $  15. 

5.  750  bbl.  pork  @$  15.12^. 

6.  24  doz.  cans  tomatoes  at  1 1.12J. 

7.  90  doz.  cans  peas  @  $  1.37^. 

8.  35  boxes  figs  @  $  1.62|. 

9.  14911b.  tea  @33|^. 

10.  450  sugar-cured  hams,  5400  lb.,  @  12J^. 

11.  348  boxes  oranges  @  $  1.75. 

12.  117  bbl.  flour®  $6.75. 

13.  16  hams,  195  lb.,  @  16f  ^. 

14.  48  doz.  cans  of  tomatoes  @  75  ^. 

15.  108  bbl.  N.Y.  buckwheat  flour  @  $4.25. 

16.  125  doz.  cans  tomatoes  @  $2.66|. 

17.  96  doz.  cans  peas  @  $  1.37^. 


FRACTIONAL  PARTS  43 

18.  6  tierces  refined  lard,  2096  lb.,  @S^^. 

19.  330  doz.  jars  of  mustard  @  87^)?. 

20.  368  1b.  coif ee@  20  j«. 

21.  178  bbl.  beef  @  120. 

33.  Cost  of  Goods  Sold  by  the  Hundred.      Freight  tariffs  ^ 
are  usually  based  on  one  hundred  pounds.     Live  stock  is 
quoted  by  the  hundredweight,  and  many  other  articles  are 
sold  in  lots  of  one  hundred.     Since  cents  are  hundredths 

of  a  dollar,  goods  sold  hy  the  hundred  will  cost  as  many  cents 
per  unit  as  dollars  per  hundred.  Thus  $5  per  cwt.  is  5^  per 
pound,  and  f  3.75  per  C  is  f  .0375  per  pound. 

Find  Cost  of: 

1.  Freight  charges  on  4230  lb.  of  merchandise  at  $1.40  per  cwt. 
1.014  X  4230  =  159.22.     (|4.23  x  14  =  $59.22.) 

2.  1280  lb.  nails  @  32^  per  cwt. 

3.  2963  lb.  rock  salt  @  $1.20  per  cwt. 

4.  2974  lb.  fence  wire  @  $4.50  per  cwt. 

5.  1280  posts,  split,  @  $18  per  C. 

6.  375  lb.  lead  @  $4.75  per  cwt. 

7.  5  cattle,  averaging  925  lb.,  @  $4.50  per  cwt. 

8.  11,580  posts,  round,  @  $25  per  C. 

9.  9850  lb.  pork  @  $7.50  per  cwt. 

10.  Freight  on  a  carload  of  grain,  46,500  lb.,  @  38)^  per  cwt. 

11.  1378  1b.  poultry  @  $6.50  per  cwt. 

12.  1865  lb.  beef  @  $14  per  cwt. 

13.  975  lb.  bran  @  90^  per  cwt. 

34.  Goods  Sold  by  the  Thousand.  Brick,  lumber,  shingles, 
and  many  other  articles  are  sold  by  the  thousand.  Gas  is 
sold  by  the  thousand  cubic  feet.  The  amount  consumed  is 
shown  by  a  meter,  upon  the  face  of  which  are  usually  three 
dials.  The  dial  to  the  right  shows  hundreds  of  cubic  feet, 
that  in  the  middle  shows  thousands,  and  that  to  the  left 


44  ELEMENTS  OF  BUSINESS  ARITHMETIC 

shows  tens  of  thousands.  Reading  the  last  figure  passed  by 
the  pointers,  and  going  from  left  dial  to  right,  the  number  of 
hundreds  of  cubic  feet  consumed  is  shown.  Thus,  the  ac- 
companying illustration  reads  323  hundreds  or  32,300  cubic 


A  Gas  Meter 

feet.  The  reading  for  the  preceding  period  is  shown  by  the 
dotted  lines.  The  reading  for  the  preceding  period  is  sub- 
tracted from  the  present  reading  to  find  amount  of  gas 
consumed. 

A  mill  being  j-^q-q  of  a  dollar,  articles  sold  hy  the  thousand 
are  as  many  mills  per  article  as  dollars  per  thousand.  Thus, 
137  per  M  is  37  mills  per  pound  or  foot,  or  126.50  per  M  is 
26.5  mills  (f  .0265)  each. 

Find  Cost  of: 

1.  1625  ft.  oak  lumber  @  $36  per  M. 

One  foot  would  cost  36  mills  (1.036),  and  1625  ft.  would  cost  $.036  x 
1625  ($1,625  X  36)  or  $58.50. 

2.  20,408  ft.  flooring  @  $17.50  per  M. 

3.  14,450  ft.  2  X  6-15  @  $15.50  per  M. 

4.  10,458  ft.  6  X  6-18  @  $22.50  per  M. 

5.  10,448  ft.  sheathing©  $12.75  per  M. 

6.  612  ft.  pine  lumber  @  $15  per  M. 

7.  456  ft.  spruce  @  $12.50  per  M. 

8.  7750  shingles  @  $5.25  per  M. 

9.  The  meter  readings  for  gas  consumed  in  a  residence  for  six 
months  were  as  follows:  Sept.  1,  28,400;  Oct.  1,  31,000;  Nov.  1, 
34,400;    Dec.  1,  37,600 ;   Jan.  1,  40,700 ;    Feb.  1,  43,200;    Mar.  1,  45,100. 


FRACTIONAL  PARTS  45 

At  90  ^  a  thousand,  what  was  the  total  of  the  monthly  gas  bills  for  the 
six  months? 

10.  A  contractor  bought  material  for  a  building  as  follows :  567,800 
brick  @  $6.75  ;  35,657  ft.  matched  pine  @  $  20 ;  14,720  ft.  hemlock  @  $12 ; 
4680  ft.  walnut  @  $45 ;  75,250  shingles  @  |4.75.    What  was  the  total  cost  ? 

11.  If  there  is  a  gas  meter  in  the  school  building,  read  it  from  month 
to  month  and  compute  the  cost  of  gas.  Also  read  the  meters  in  your 
home  and  compute  the  cost  of  gas. 

35.  Goods  Sold  by  the  Ton.  Coal,  hay,  and  other  articles 
are  sold  by  the  ton.  If  coal  is  quoted  at  $  6.50  per  ton,  it  is 
$3.25  per  1000  pounds,  and  3.25  mills  (1.00325)  per  pound. 
In  other  words,  goods  sold  hy  the  ton  are  one-half  as  many 
mills  per  pound  as  dollars  per  ton.  Thus,  $3.80  per  ton  is 
1.9  mill  (1.0019)  per  pound. 

PROBLEMS 

1.  3640  lb.  hay  @  $8.50  per  T. 

1  of  $8.50  =  $4.25  per  M.,  or  4.25  mills  per  lb.,  $.00425  x  3640=$15.47. 

2.  42,300  lb.  hay  @  $9  per  T. 

3.  2860  lb.  soft  coal  @  $3.75  per  T. 

4.  84,375  lb.  steel  rails  @  $20  per  T. 

5.  225,780  lb.  ore  @  $25  per  T. 

6.  3500  lb.  fertilizer  @  $22  per  T. 

7.  4525  lb.  anthracite  coal  @  $8.75  per  T. 

8.  38,960  lb.  salt  @  $3.35  per  T. 

9.  265,700  lb.  clover  hay  @  $  6.20  per  T. 

10.  What  will  be  the  cost  of  the  freight  on  5  cars  of  coal  at  $  2.75  per 
ton,  the  cars  weighing  respectively :  87,560,  75,605,  54,780,  85,670,  and 
70,840  pounds  net? 

11.  Find  the  Value  of  Each: 


Article 

Gross  Weight 

Tabb 

Priob 

A  load  of  hay 

3450  lb. 

1256  lb. 

$8.75  per  T. 

A  load  of  straw 

2975  lb. 

856  lb. 

$2.60  per  T. 

A  load  of  coal 

5475  lb. 

1680  lb. 

$7.50  per  T. 

A  load  of  beets 

3475  lb. 

1240  lb. 

$12..50  per  T. 

A  load  of  stone 

4250  lb. 

1250  lb. 

$14.50  per  T. 

46 


ELEMENTS  OF  BUSINESS  ARITHMETIC 


36.  Invoicing,  Bills,  and  Accounts.  When  goods  are 
shipped,  or  sold  on  account,  an  invoice  or  bill  is  mailed,  or 
sent  with  the  goods.  A  detailed  statement  of  the  amount, 
kind,  and  prices  of  the  goods,  together  with  the  names  of  the 
parties  to  the  transaction,  term  of  credit,  condition  of  sale, 
discount  allowed,  etc.,  is  called  either  a  bill  or  invoice.  The 
term  bill  is  applied  particularly  to  a  statement  of  goods 
bought,  of  services  rendered,  or  of  work  performed.  Prompt- 
ness in  billing  goods  shipped,  and  accuracy  in  computing  the 
amount  of  the  bills,  are  business  essentials. 

Formerly,  the  term  "  invoice  "  was  used  when  goods  were 
shipped  on  consignment  only,  but  it  is  now  often  used  inter- 
changeably with  "bill."  A  clause  on  the  invoice,  stating  that 
the  goods  remain  the  property  of  the  consignor  until  paid  for, 
makes  the  sending  them  out  a  consignment  rather  than  a  sale. 

At  stated  periods,  usually  the  first  of  each  month,  a  state- 
ment of  account  is  sent  to  debtors.  A  statement  merely 
gives  the  amount  of  bills  previously  rendered,  and  credits, 
with  their  dates.  Its  chief  purpose  is  to  remind  the  debtor 
of  the  debt,  but  it  is  also  of  assistance  in  checking  errors  in 
accounts. 

PROBLEMS 

Rule  forms  and  copy  the  following  invoices,  filling  in  the  missing 
amounts. 

Philadelphia,  Pa.,  Jan.  19, 19— 
1.  The  Amos  King  Co., 
Baltimore,  Md. 


Bought  of  JOHN 

WANAMAKER 

Terms :  "Net  30  days 

1310 

10 

Roll  Top  Office  Desks, 

143.75 

X1338 

15 

Typewriter  Desks, 

4.50 

1317 

20 

Office  Tables, 

7.00 

1238 

10 

Office  Chairs, 

6.75 

Note.  —  The  figures  to  the  left  of  the  ruling  show  the  catalogue  numbers. 


FRACTIONAL  PARTS 


47 


2. 


PitUburg,  Pa^      /i^r^yy 

/. 

_19= 

- 

Xh^-iT-i^^^^T'T^^y/^-^^/^     Uy.    - 

in.ccmitwhh  A.  D.  HIRSCH  &  COMPANY 

^^. 

/^Tjiyf^. 

/ 

-^.^rr^  a^^.^<h-fy^yy7^yA:y^^.^^^^^  ^^,^ 

.^r 

^-r^ 

/^ 

..  %^^.^ 

/ 

i.^ 

,^,6-r. 

7.^; 

~^^y 

X— ' 

/^ 

■"■?W<?/  .^^^r^r^A^^^ 

/  /O  o 

or? 

?-^/ 

,?/^/^ 

(L^ 

' 

Chicago,  111.,  May  6, 19— 

3.  Mr.  S.  a.  Daubb, 

Evanston,  111. 

Bought  of 

Terms :   30  da.  2  %  10  da. 

1690  ft.  N.C.  Ceiling $18.50 

20,165  ft.  Flooring 28.50 

3520  ft.  Studding 10.80 

12,500  Shingles 5.75 

4.  Make  out  bill  and  find  the  amount  of  the  following  invoice  of 
goods,  shipped  to  Messrs.  Chase  and  Witherspoon,  Richmond,  Ind.,  by 
The  Sprague  Grocery  Co.,  Chicago,  111.,  May  1,  19 — ;  terms:  60  da.  net, 
3%  10  da. :  20  hf.  chests  Japan  Tea,  1200 #,  @  23j2!;  20  hf.  chests  Oolong 
Tea,  1000 #,  @  48^;  16  cases  Ceylon  Tea,  800  #,  @  35^;  10  bags  Mocha 
Coffee,  1500  #,  @  24^;  20  bags  Java  Coifee,  1500  #,  @  25^. 

Find  the  amount  of  the  following  invoices : 

5.  N.  E.  Nash  &  Co.,  Chicago,  sold  Young  &  Hooper,  Ft.  Wayne,  Ind., 
135  bbl.  Mess  Pork  @  $11.50;  40  bbl.  Mess  Beef  @  $11.75;  SOtigtcfiS;,^^^^ 
R'fd  Lard,  10,500  #,  @^,f\  450  Sugar-cured  Hams,  1500  #,  (^,J:^^RY^ 


48 


ELEMENTS  OF  BUSINESS  ARITHMETIC 


6.  A.  D.  Cunningham  bought  of  C.  E.  Kinsley  on  account  30  da., 
24  pr.  Congress  Shoes  @  .|1.60;  25  pr.  Men's  Heavy  Shoes  @  $1.75; 
22  pr.  Ladies'  French  Kid  Shoes  @  $3.25 ;  12  pr.  Boys'  Shoes  @  $.80;  14 
pr.  Infants'  Shoes  @  $.90;  22  pr.  Calf  Boots  @  $2.25.  Make  out  the 
bill,  supplying  dates  and  places,  and  take  off  |  for  cash. 

7.  5  Brass  Bedsteads  No.  200  @  $30;  6  rolls  Wilton  Carpet,  280  yd., 
@  80^;  4  Hair  Mattresses,  Style  BB,  @  $12.50;  16  rolls  Ingrain  Carpet, 
1256  yd.,  @  40  ^ ;  8  rolls  Moquette,  432  yd.,  @  85  )^ ;  12  pr.  Portieres  @  $  8 ; 
5  Antique  Oak  Bedsteads  @  $15.20;  discount  ^V 


New  York,  N.  Y.,     ~^6/l^ 

7-f7, 

_19^= 

— 

/t^^f^^^^^--p-?^.-^     /^^.^ 

^'^  R  H.  MACY  ca.  COMPANY 

Terms;    ^.^.d^-A—                                                           255  Broadway 

S 

^2l^^^<P^N-^.^7^^d^                                    7Z.J/^      Z3  4 

/^ 

„r^ 

Z 

>^v^^^^^y^-^>^            /r.  ..     ^r. 

,3  7 

.To 

7- 

r.^^^^ — '/.^f;-?^^f:7jt?h-^^^~                  tc^^    /.  ^' 

7 

?  0 

,? 

JJJ.   .J^^^/?^.w^J^^Z^.^  ""   ^.^ 

/a 

,r^ 

^^^^^^^A-^..,:^Sr/^^^i^^^^.^^ 

-^^<A^^^^.  ^^^ 

^y^^-i^-/y/7?7 

Render  the  following  statements: 

9.   Jan.  31,  19 — ,  the  debits  and  credits  of  John  C.  Scott  in  account 
with  the  Simmons  Hardware  Co.,  of  St.  Louis,  Mo.,  were  as  follows : 

Debits :  Jan.  1,    To  account  rendered,  $  157.25 
Jan.  7,    To  Mdse.  $32 
Jan.  12,  ToMdse.  $345 

Credits  :  Jan.  5,  By  cash,  $  150 

Jan.  10,  By  20-day  note,  $60 

10.   Oct.  31,  19—,  the  debits  and  credits  of  W.  C.  McClure  with  Lord 
&  Taylor,  New  York  City,  were : 


FRACTIONAL  PARTS 


49 


Debits :  Oct.  1,    To  account  rendered,  $86.25 

Oct.  10,  To  Mdse.,  $  165 

Oct.  19,  ToMdse.,  $425 

Oct.  29,  To  Mdse.,  $  136.50 
Credits :  Oct.  5,    By  cash,  $  80 

Oct.  20,  By  note  for  60  days  for  balance  due 

11.   On  Nov.  30,  19 — ,  the  account  of  Wm.  E.  Bacon  with  American 
Prism  Co.,  Cincinnati,  O.,  was  as  follows : 

Debits :  Nov.  2,    To  Mdse.,  $  109.60 
Nov.  8,    To  Mdse.,  $67.80 
Nov.  21,  To  Mdse.,  $240 
Credits :  Nov.  15,  By  cash,  $  80 

Nov.  20,  By  10-day  note,  $75 
Nov.  29,  By  cash,  $  200 

37.    Pay  Rolls. 


PAY  ROLLi 

For  the  week  ending  _ 

19 

N. 

Number  of  H<M>r.  Work 
E.ch  D., 

Told 
No.  oi 
Hours 

per 
Hour 

Toial 
W^ei 

Rco-rk. 

M 

T 

w. 

T. 

F 

s. 

.  ^-^-^2^.    ^ 

r 

f 

r 

^ 

^ 

^f 

X^l 

/? 

2  ^^-^...7>.y 

<^'/^ 

T'A 

r'/, 

ur'A 

.^09 

/// 

<:< 

3  iCP'OfOh.^J^ 

^ 

■r 

r 

^'/„ 

r 

7//-^ 

4    ^(;^.-^i&.-r>nf.A^ 

r 

r 

r 

)(• 

f 

x.r<^ 

s  -^y/Jh/h.^^^ 

,P 

f 

-r 

f 

-r'/, 

r 

3S<^ 

6    ^^^^^.A,-^^ 

^ 

. 

e% 

^ 

f 

r 

'C'A 

r     'Ofy^%'^.y.^^ 

^ 

r 

r'A 

^ 

r'A 

rf 

yp/. 

8    0-^::^:^y.^ 

^ 

r 

/f 

f 

^'/, 

ZJ-</' 

9  ""0^:^^^^^... 

a- 

a 

rf 

^ 

n"/. 

^•7^M 

10        (2^^-\d^^.y^^y 

jp 

^ 

^'A 

?f7|/. 

^        J       J^  - 

_ 



Copy  the  above  pay  roll  and  fill  out  the  blank  spaces  as 
shown  in  Nos.  1  and  2. 

Some  firms  use  checks  in  paying  off  employes,  but  many 
find  it  more  convenient  to  pay  in  cash  by  the  envelope 
system.  To  pay  off  men  by  the  envelope  system,  it  is  neces- 
sary to  find  first  the  amount  of  money  needed,  then  the  bills 
and  fractional  currency  needed.     To  do  this,  a  blank  called 


50 


ELEMENTS  OF  BUSINESS  ARITHMETIC 


a  change  memorandum^  similar  to  the  one  shown  below,  is 
generally  used. 

Fill  out  the  accompanying  change  memorandum  for  the 
pay  roll  given  above.     Make  totals  and  check  results. 


No. 

Bills 

Coins 

$20 

$10 

$5 

$2 

$1 

50  j^ 

25^ 

nf 

5^ 

1^ 

1 

2 

3 
4 
5 
6 
7 
8 
9 
10 

1 
1 

1 

2 

1 

1 

FIRST  NATIONAL   BANK 

Union  City,  Mo. 

Pay-roll  Memorandum 

Union  City  Mfg.  Co.  require  the 

following : 

Pennies      ...     8            .08 

Nickles 

6            .30 

Dimes   . 

7            .70 

Quarters 

8          2.00 

Halves  . 

6          3.00 

Dollars 

4          4.00 

2'8      .     . 

7        14.00 

5's     .     . 

3        15.00 

lO's  .     . 

9        90.00 

20'8  .     . 

1        20.00 

149.08 

When  the  amount  of  the  pay 
roll  and  the  necessary  bills  and 
fractional  currency  have  been 
determined,  a  pay-roll  memo- 
randum^ similar  to  the  accom- 
panying form,  is  taken  to  the 
bank  and  the  necessary  amount 
of  each  denomination  of  money 
secured.  The  pay-roll  memo- 
randum should  foot  the  same  as 
the  pay-roll  book. 

Make  out  a  pay-roll  memo- 
randum for  the  preceding  pay 
roll,  using  the  ajCcompanying 
form  as  a  model. 


FRACTIONAL  PARTS  51 

PROBLEMS 

1.  A  man  worked  9  months,  26  days  per  month,  and  received  $  912.60. 
What  was  his  daily  wage  ? 

2.  I  hold  $  2000  worth  of  stock  in  a  company  capitalized  at  $  36,000. 
A  dividend  of  $  4200  is  declared.     What  is  my  share  ? 

3.  I  have  $1200  on  interest  at  5%.  What  does  it  yield  annually? 
(5%=^.) 

4.  Load  deliveries  of  coal  at  a  school  building  for  the  month  of  Feb- 
ruary were  as  follows:  4500,  4650,  3850,  4000,  3675,  3800,  4200,  4350, 
4500,  4100,  4350,  4400,  4050,  3900,  3600,  3550,  and  3850  lb.  net.  Find 
the  total  delivery,  the  average  weight  per  load,  the  total  cost  at  $  4.20 
per  ton,  and  the  average  cost  per  load. 

5.  A  society,  after  raising  $  17,675  by  subscription,  erected  a  building 
that  cost  $  22,500.  How  much  did  each  of  its  25  members  have  to  pay 
on  a  per  capita  assessment  ? 

6.  In  a  woolen  factory  there  are  48  looms.  If  in  208  days  they  wove 
269,568  yd.  of  cloth,  how  much  was  woven  on  each  loom  ?  How  much 
on  all  in  one  day  ?    On  each  in  one  day  ? 

7.  A  shoe  dealer  bouoht  35  cases  of  boots,  containing  12  pairs  each. 
For  the  lot  he  paid  |2100.  How  much  did  they  cost  per  case?  Per 
pair? 

8.  The  entire  cost  of  constructing  a  railroad  25  miles  long  was 
$  1,736,075.     What  was  the  cost  per  mile  ? 

9.  The  population  of  Rhode  Island  in  1900  was  428,556.  The  area  is 
1250  sq.  mi.     What  was  the  number  of  inhabitants  to  the  square  mile  ? 

10.  A  schoolroom  containing  72,500  cu.  ft.  of  air  is  occupied  by  125 
pupils.  How  much  air  is  that  to  each  pupil?  If  the  air  is  completely 
changed  every  10  minutes,  how  much  fresh  air  is  provided  for  each  J)upil 
per  hour? 

11.  A  county  containing  488,000  acres  is  divided  into  27  townships 
of  equal  area.     How  many  acres  in  each  township  ? 

12.  The  population  of  the  United  States  in  1900  was  74,627,907.  The 
number  of  congressmen  was  357.  What  was  the  average  number  of 
persons  represented  by  each  congressman  ? 

13.  An  agent  earns  monthly  commission  as  follows  :  $  75,  $  92,  $  83, 
$  66,  $  55,  $  28,  ^  82,  1 160,  $  32,  $  280,  $  175,  $  196,  and  $  215.  What  is 
his  average  commission  per  month  ? 


52 


ELEMENTS  OF  BUSINESS  ARITHMETIC 


14.  In  one  year  the  national  banks  of  New  York  City  loaned 
$  236,327,598.     What  was  the  average  weekly  loan  ? 

15.  The  total  cost  of  building  the  Union  Pacific  Railway  was 
$  18,941,400.    The  road  is  1177  miles  long.    What  was  the  cost  per  mile  ? 

16.  A  man  bought  a  house  and  lot  for  $6107.  He  paid  $1392  down 
and  agreed  to  pay  the  balance  in  115  monthly  payments.  What  was  the 
amount  of  each  payment  ? 

17.  If  a  gas  burner  consumes  1100  cu.  ft.  of  gas  in  160  hours  at  a 
cost  of  $1.42  per  thousand,  what  is  the  cost  per  hour?  How  many 
cubic  feet  does  it  burn  in  one  hour  ? 

18.  How  much  will  it  cost  to  ship  a  carload  of  corn  containing 
45,000  lb.  from  Crete,  Neb.,  to  Chicago,  if  the  freight  rate  is  6^  per  hun- 
dred pounds?     What  would  be  the  cost  per  bushel  of  56  lb.  ? 

19.  The  average  wages  of  a  steel  mill  employing  4500  men  is  $  3.50 
per  day.  If  a  reduction  of  ^  is  made  in  wages,  how  much  is  the  com- 
pany's pay  roll  reduced? 

20.  Find  the  amount  of  the  following  invoice  of  goods  : 

10  pc.  sateen,  55|,  51,  50|,  52,  57,  54^,  50,  56-1,  5.31,  51  @  6|  ^ ;  11  pc. 
flannel,  62^,  60,  65,  67|-,  65,  64,  63^,  67,  78,  62|,  63  @  6|  ^ ;  15  pc.  ging- 
ham, 50,  51,  52,  54^,  56,  57,  58|,  54|,  59,  60,  61,  56i,  52^,  53|,  54  @  12^;^. 

21.  Find  the  amount  of  the  following  pay  roll  (a)  on  the  basis  of  an 
8-hour  day  ;  (6)  on  the  basis  of  a  10-hour  day : 


Name 

MON. 

TuES. 

Wed. 

Tm-RS. 

Fri. 

Sat. 

Eate 

Lawrence  Dodd 

7 

8 

10 

9 

8 

11 

$3.00 

Henry  Wahl 

8 

8 

8 

8 

8 

10 

2.50 

James  Ringo 

6 

7 

5 

8 

9 

1 

3.50 

Leroy  Hines 

9 

6 

8 

10 

12 

12 

3.00 

Thomas  Cox 

8 

8 

8 

9 

9 

10 

1.75 

Amos  Zeering 

9 

9 

9 

9 

9 

8 

2.50 

Adam  Frick 

8 

8 

8 

8 

8 

8 

3.00 

Warren  Hastings 

9 

10 

10 

10 

11 

12 

2.25 

Gordon  Bang 

7 

6 

5 

6 

8 

8 

3.00 

Emil  Ehman 

8 

8 

9 

5 

6 

5 

4.00 

22.   Make  pay-roll  memorandum  and  change  memorandum  for  above 
problem. 


FRACTIONS 

38.  Definitions.  By  fraction  is  meant  one  or  more  of  the 
equal  parts  of  some  quantity.  Tlie  size  of  a  part  is  indicated 
by  the  denominator  in  showing  the  number  of  parts  into 
which  the  wliole  is  divided.  Inasmuch  as  fourths  and  halves 
are  as  truly  different  things  as  bushels  and  pecks,  it  follows 
that  the  use  of  the  denominator  is  to  denominate  or  name 
the  kind  of  parts.  The  number  of  parts  used  is  indicated  by 
the  numerator  (numberer).  Thus,  in  |  and  |,  and  in  5  bu. 
and  3  pk.,  the  denominators  8  and  7  and  the  words  bushels 
and  pecks  indicate  the  kind  of  parts  or  units,  while  5  and  3, 
in  each  case,  show  the  number  of  those  units.  The  numerator 
and  denominator  are  called  the  terms  of  the  fraction. 

By  common  or  vulgar  fractions  are  meant  fractions  of 
ordinary  kind,  i.e.  fractions  expressed  by  a  numerator  written 
above  a  line  and  a  denominator  written  below  the  line.  A 
common  fraction  whose  denominator  is  1,  with  a  number  of 
ciphers  annexed,  is  called  a  decimal  fraction,  because  its 
denominator  expresses  a  decimal  denomination  (Sec.  16). 
Such  a  fraction  may  be  written  at  pleasure  as  a  decimal 
fraction  or  as  a  decimal.     (Sec.  17.) 

A  fraction  which  represents  a  part  of  a  quantity  less 
than  unity  is  called  a  proper  fraction;  if  it  is  equal  to  or. 
greater  than  unity,  it  is  improper.  The  numerator  of  a 
proper  fraction  is  numerically  less  than  the  denominator; 
the  numerator  of  an  improper  fraction  is  equal  to  or  greater 
than  the  denominator.  A  mixed  number  contains  a  whole 
number  and  a  fraction. 

53 


54  ELEMENTS  OF  BUSINESS  ARITHMETIC 

Business  computations  require  constant  use  of  fractions. 
A  practical  knowledge  of  fractions,  however,  necessitates 
only  a  thorough  mastery  of  operations  with  the  simpler 
fractions,  and  skill  in  their  use.  Little  attention  need  be 
given  to  fractions  having  larger  terms. 

39.  Reduction  of  Fractions.  The  same  fraction  may  be 
expressed  in  different  denominations.  Thus,  ^  may  be 
written  as  |  or  |. 

A  change  in  denomination  requires  a  change  in  the  de- 
nominator. If  the  value  of  the  fraction  is  not  to  be  changed, 
then  the  numerator  must  also  be  changed  and  in  exactly  the 
same  ratio.  In  other  words,  the  terms  of  a  fraction  may  he 
multiplied  or  divided  hy  the  same  number  without  changing  the 
value  of  the  fraction. 

When  the  change  results  in  terms  of  less  numerical  value, 
the  fraction  has  been  simplified  or  reduced  to  lower  terms. 

It  should  be  remembered  that  one  may  add,  subtract,  or 
divide  like  quantities  only.  Three  quarts  may  not  be  added 
to  five  pints  until  they  are  changed  to  the  same  denomina- 
tion. Neither  is  it  possible  to  add  |  and  |  or  to  find  how 
many  |'s  there  are  in  |  without  reducing  to  like  denomina- 
tion. In  division  of  fractions  this  step  is  obscured  by  the 
commonly  used  process  of  inversion,  but  it  is  nevertheless 
performed.  The  recognition  of  these  facts  and  their  con- 
stant application  will  do  much  to  simplify  the  whole  subject 
of  fractions  and  give  facility  in  computations  involving  them. 

Simplify  : 

•       1.  m  7.   Itl  13.  ^\%  19.  ^^  25.  ^^, 

2.  tVt  8.   ^%%  14.  IM  20.  lU  26.  i|f 

3.  H  9-   n  15-  .Vr           .  21.  m  27.  f|« 

4.  iU  10.   U  16.  m  22.  ^^\  28.  lit 

5.  M  11.   U  17.  i^,  23.  ^^^  29.  Hf 

6.  j^^  12.   11}  18.  ,^  24.  ^f  30.  ill 


FRACTIONS  55 

40.  Reduction  to  Like  Denomination  by  Inspection.  Chang- 
ing fractions  to  fractions  of  like  denomination  is  called 
reducing  to  a  common  denominator.  Fractions  of  like  de- 
nominators are  similar  fractions.  The  practice  problems 
here  given  deal  with  the  more  commonly  used  fractions  and 
are  intended  to  develop  the  power  of  readily  thinking  one 
fraction  in  terms  of  another;  e.g.  |,  l,  ^,  as  -f^,  -f-^^  and  -f^. 
This  is  one  of  the  operations  in  fractions  most  frequently 
needed. 

DRILLS 

1.  I  =  how  many  4ths,  Gths,  lOths,  16ths,  18ths,  24tlis,  28ths,  36ths, 
72ds,  84ths,  60ths,  56ths,  42ds,  oOths,  44ths,  70ths? 

2.  i  =  how  many  9ths,  15ths,  21sts,  24ths,  27ths,  36ths,  45ths,  48ths, 
51sts,  54ths,  63ds,  72ds? 

3.  ^  =  how  many  8ths,  12ths,  16ths,  24ths,  28ths,  32ds,  36ths,  48ths, 
52ds,  56ths,  60ths  ? 

4.  ^  =  how  many  15ths,  25ths,  30ths,  35ths,  45ths,  50ths,  55ths,  65ths  ? 

5.  ^  =  how    many   12ths,    18ths,   24ths,   30ths,   36ths,  42ds,   48ths, 
54ths,  60ths,  66ths,  72ds? 

6.  i  ==  how    many   16ths,   24ths,   32ds,  40ths,   48ths,   56ths,   64ths, 
72ds? 

7.  f  =  how    many    14ths,    21sts,  28ths,   35ths,  42ds,  49ths,   56ths, 
63ds,  70ths? 

8.  ^-^  =  how  many  20ths,  30ths,  40ths,  50ths,  60ths,  70ths? 

9.  ^  =  how  many  ISths,  27ths,  36ths,  45ths,  54ths,  63ds? 

10.  fr  =  how  many  22ds,  33ds,  44ths,  55ths,  66ths,  77ths? 

11.  j\  =  how  many  24ths,  36ths,  48ths,  60ths,  72ds  ? 

12.  fj  =  how  many  26ths,  39ths,  52ds,  65ths,  78ths? 

13.  j\  =  how  many  28ths,  42ds,  56ths,  70ths? 

14.  ^3  =  how  many  30ths,  45ths,  60ths,  75ths? 

15.  2V  =  tow  many  40ths,  60ths,  80ths? 

16.  From  comparison  with  their  equivalents,  as  required  in  the  above 
problems,  to  what  could  one  change  or  reduce  halves,  fourths,  and  eighths 
so  that  they  would  be  of  the  same  units  or  kinds  of  parts  ?    What  halves, 


56  ELEMENTS  OF  BUSINESS  ARITHMETIC 

fourths,  eighths,  and  sixths  ?    What  halves,  fourths,  eighths,  sixths,  and 
twelfths? 

17.   Reduce  thirds,  sixths,  and  ninths  to  common  denominators.     3ds, 
6ths,  12ths,  and  15ths.     3ds,  6ths,  9ths,  and  12ths. 

41.    Operations  with  Easily  Reducible  Fractions. 

1.  $5  + S3 +12  =  $10.     23bu. +  15bu.4-12bu.  =  50bu. 

Likewise  ^  +  ^  +  1  +  ^=  2^3.^  or  5|. 

2.  $56-112  =  144.     24  pk.  -  19  pk.  =  5  pk. 

Likewise  |  —  -|  =  |,  or  -|. 

3.  181^19  =  9.     72bu. -f-6bu.  =  12. 

Likewise  §J  -5-  ^\  =  24  --  3,  or  8. 
Take  note  that  in  the  division  of  fractions  having  the  same 
denominator^  the  denominators  are  disregarded^  and  the  quotient 
of  the  fractions  is  the  quotient  of  the  numerators. 

Reduce  to  Similar  Fractions  by  Inspection  and  Add: 


1.  i  1,  h  h  ^ 

5. 

h  h  t\,  t\ 

2.  h  h  A, 

It\ 

6. 

4J  A)  TSf  ^i 

3.    1,1,  A, 

hj\ 

7. 

h  ^%  h  If 

4.   f,>,T\, 

A 

8. 

2)    6J    3,    9 

Subtract 

: 

1.  A-^ 

5. 

f- 

■f 

• 

9. 

15^ 

-n 

2.    f-f 

6. 

tV- 

-1% 

10. 

16|. 

-5| 

3.   l-i 

7. 

j% 

-z\ 

11. 

163- 

-98f 

4.    i-T^B 

8. 

i- 

-ih 

12. 

.85- 

-14| 

Divide  : 

Note.  —  All  easily  reducible  fractions  are  more  quickly  and  accurately 
divided  by  reducing  to  similar  fractions.  In  practice,  write  the  numera- 
tors only. 


i-^T%=^%-^ 

•if  = 

=  T^ir 

1. 

f^f 

5.    U-^ii^ 

9. 

16^  --  4  J 

2. 

^^i 

6.    !|-^t\ 

10. 

f-f-41 

3. 

tV-A 

7.   1^1 

11. 

61-^  H 

4. 

f-^^ 

8.  A^f 

12. 

t¥t-^t'j 

2)^-9-^-j[^-7-14-^-12 
3)       9 7  6 

3  7  2 


FRACTIONS  57 

42.  Addition  and  Subtraction  through  finding  the  Least 
Common  Multiple.  For  fractions  not  easily  reducible  to  a 
common  denominator  by  inspection,  the  method  of  finding 
the  Least  Common  Multiple  of  all  the  denominators  is  used 
in  order  to  know  the  least  denomination  to  which  all  of  the 
fractions  may  be  reduced. 

Problem.  —Add  f,  ^,  8f ,  f^,  5f,  ^^,  7|,  f|-. 
In  order  to  find  the  Least  Common  Multiple,  it  is  usual  to 
write  all  the  denominators,  then  strike  out  any  one  that  is 

repeated  or  that  is  a  factor 
of  any  other;  as,  3,  6,  14, 
7,  and  2.  The  remaining 
numbers  are  then  divided 
2x3x3x7  X  2  =  252,  L.C.M.     y^^  ^„y  divisor  of  two   or 

more  of  them,  bringing  down  any  number  not  divisible. 
This  is  repeated,  if  necessary,  until  no  two  of  the  numbers 
remaining  have  a  common  divisor.  The  product  of  the 
divisors  and  all  of  the  numbers  remaining  will  be  the  Least 
Common  Multiple  of  the  given  numbers. 

43.  To  reduce  to  a  Common  Denominator.  Since  252  is  a 
multiple  of  each  of  the  denominators  of  the  fractions  given 
in  the  preceding  section,  each  fraction  may  now  be  reduced 

168  to  an  equivalent  fraction  having  252  for  its 

112  denominator.     To  do  this,  it  is  well  to  arrange 

210  the   fractions   vertically,  placing    the   desired 

90  denominator  below  a  line  and  directly  beneath 

216  the  fractions.     To  the  right  of  a  line  drawn 

162  vertically,  and  opposite  each  fraction,  is  written 

126  the  numerator  of  the  equivalent  fraction.    The 

231  method  used  in  finding  this  equivalent  fraction 


3 

of 


71 
11 

12 


252jl816(5  is  the  same  as  by  inspection.     (Sec.  40.) 

1260  i  =  ^AO'«-*of  |f|),andf  =  i|f.     J=AV 

ii^       and|  =  Ji|.     H^jandf=e|,etc.     Add- 


58  ELEMENTS  OF  BUSINESS  ARITHMETIC 


20 


ing  the  numerators,  the  sum  of  the  fractions 
is  seen  to  be  ^gV^  ^^  ^  ^^^  /A*  Added 
to  20,  the  sum  of  the  whole  numbers,  the 
sum  required  is  found  to  be  25  and  ^-^. 


PROBLEMS 

1.  Add  7,  8f,  9i^,  6j\,  5t\,  and  3^. 

2.  Add  i  I,  5|,  12^23,  and  xV- 

3.  Subtract  |f  from  7||^. 

4.  Subtract  yff ^  from  256ff. 

5.  From  If  +  llf  take  12\  -  9|. 

6.  From  the  sum  of  75|  and  94f  take  the  sum  of  36^^  and  24|. 

44.   Multiplication  of  Fractions  and  Whole  Numbers. 

(a)  Five  times  3  bu.  =  ?  Five  times  3  fourths  =  ?  Five 
times  f=?  |x5=?  How  many  wholes  ?  (Use  multi- 
plication sign  correctly.     Sec.  13.) 

5  times  |  =  J^  or  3|. 

(6)  5  X  f  =  ?  This  may  also  be  written,  |  of  5  =  ?  In 
the  latter  form  it  is  recognized  as  finding  the  value  of  frac- 
tional parts.     See  Chapter  IV. 

I  of  5  =  f  X  3  =  -I45.  or  3f . 

A  fraction  may  be  multiplied  hy  multiplying  its  numerator. 
Dividing  the  denominator  also  multiplies  the  fraction  by 
increasing  the  size  of  the  denomination.  A  fraction  may  be 
used  as  a  multiplier  hy  dividing  hy  its  denominator  and  multi- 
plying the  quotient  hy  the  numerator.  This  process  may  some- 
times be  shortened  by  first  multiplying  by  the  numerator 
and  then  dividing  by  the  denominator. 

In  multiplying  mixed  numbers,  the  whole  numbers  and 
fractions  should  be  multiplied  separately  and  the  partial 
products  added.  Solve  each  problem  in  the  shortest  possible 
way. 


FRACTIONS  59 

PROBLEMS 

1.  348  X  f  =  ?  5.  284  x  3f  =  ? 

2.  tV  X  8  =  ?  6.  71f  X  3  =  ? 

3.  f  of  721  =  ?  7.  256  X  I  =  ? 

4.  12f  X  8  =  ?  8.  5xV  X  9. 

45.    Multiplication  of  Fractions. 
Solve : 

1.  f  of,^. 

Since  -J  of  -^^  is  -^^  f  is  f  ^  or  f . 

2.  Find  I  of  -|. 

Since  J  of  |-  is  -j%,  f  is  j-^^. 

Note.  —  i  x  f  =  ^\,  and  §  x  |  =  y^^ ;  that  is,  the  product 
of  the  numerators  is  divided  by  the  product  of  the  de- 
nominators. 

3.  Find  I  off. 

Since  i  of  f  (I  X  i)  =  l  |  of  f  =  f  or  i 

These  steps  may  be  performed  together  and  shortened  by 
writing  the  fractions,  canceling  the  factors  common  to  the 
numerators  and  denominators,  and  multiplying  the  remain- 
ing factors  in  the  numerators  for  the  numerator  of  the  prod- 
uct, and  the  remaining  factors  in  the  denominators  for  the 
denominator  of  the  product.     Thus, 

2x^-1   and   1  of  ?^-_22ll-12^ 

2  5         13 

4.  Multiply  5|  by  6f . 

Mixed  numbers  should  ordinarily  be  reduced  to  fractions, 
then  multiplied  as  in  3.     Thus, 

9 

.2  ^ns     17  ^  ;27     17x9     153  _  oo. 
5|x6|  =  — x^  =  -^-  =  —  or  38^. 


60  ELEMENTS  OF  BUSINESS  ARITHMETIC 

5.    Multiply  144|  by  9f . 

Reducing  a  large  number,  like  144|,  would  give  too  large 
a  numerator  to  be  handled  readily.  In  such  cases,  the  whole 
number  and  the  fraction  of  the  multiplicand  may  be  multi- 
plied separately  by  the  whole  number  and  the  fraction  of 
the  multiplier,  and  the  partial  products  added.     Thus, 


144| 

n 

A  =  (fx|) 

108      =  (f  X  144) 

3i^o=(fx9) 

1296      =(144x9) 

1407^ 

PROBLEMS 

1. 

|xf xf=? 

6.    2i  X  3|  =  ? 

2. 

|x|xf=? 

7.   f  X  If  =  ? 

3. 

f  X  A  X  ^\  =  ? 

8.  ^  xi\  =  i 

4. 

T%  X   f  X  f  X  1  =: 

=  ?                          9.  T5  of  2^  X  i  of  7} 

5.   I  of  f  of  I  of  I  =  ?  10.   9f  X  6|  =  ? 

11.  A  man  owns  f  of  a  store  and  sells  |  of  his  share  for  ^5000. 
What  was  the  store  worth,  at  this  rate  ? 

Find  cost  of : 

12.  127  bu.  wheat  at  62^^  per  bushel. 

13.  125  cords  of  wood  at  $  6|  per  cord. 

14.  17f  tons  coal  at  $  3|  per  ton. 

15.  26  bu.  clover  seed  at  $  7^  per  bushel. 

16.  22f  thousand  feet  lumber  at  $17^  per  M. 

17.  A  merchant  sold  25  yd.  satin  at  $  If  per  yard,  26|  yd.  silk  at  $2f, 
26  yd.  carpet  at  $|,  56  yd.  calico  at  3|/'.  What  was  the  amount  of  the 
sale? 

18.  Having  bought  f  of  a  farm  of  180  acres,  I  sell  f  of  my  share  at 
$40  per  acre.     How  much  do  I  receive  for  it? 

19.  A  quantity  of  provisions  will  last  25  men  12|  days.  How  long 
will  it  last  one  man  ? 


FRACTIONS  61 

46.  Division  of  Fractions.  Division  of  fractions  which  are 
not  easily  solved  by  the  method  in  Sec.  41  is  most  readily 
performed  by  the  method  of  inversion.  In  the  latter,  the 
terms  of  the  divisor  are  inverted  and  the  fractions  are  mul- 
tiplied.    Thus, 

l^i  =  ixf  =  f     (Sec.45.) 

As  stated  in  Sec.  39,  we  cannot  divide  fractions  without 
first  reducing  them  to  like  denominations.  In  the  division, 
then,  of  dissimilar  fractions  the  two  steps  are :  first,  reduc- 
tion to  like  denomination,  and  second,  the  division  of  the 
numerators.  These  two  steps  are  accomplished  simul- 
taneously by  the  inversion  of  the  terms  of  the  divisor  and 
multiplying,  thereby  shortening  the  written  process.  Thus, 
placing  the  denominator,  6,  above  the  line,  forms,  with  the 
denominator  of  the  dividend,  |  or  2.  This  is  the  number 
necessary  by  which  to  multiply  the  numerator  and  denomi- 
nator of  the  fraction  (|)  to  change  it  to  sixths,  that  it  may 
be  of  like  units  with  the  divisor  (|).  If  this  multiplication 
were  actually  made,  the  J  would  now  be  |.  The  placing  of 
the  numerator  of  the  dividend,  5,  below  the  line  indicates 
the  division  of  4  (the  numerator  of  the  dividend)  by  it,  or 
performs  the  second  step  in  the  operation.  This  gives,  as 
the  result,  -I,  the  required  result,  since  the  numerators  only 
are  to  be  divided  when  the  fractions  are  similar.  (Sec.  41.) 
The  ease  with  which  inversion  and  simple  multiplication  ac- 
complishes these  steps  as  |  x  f  =  f ,  explains  the  economy  in 
its  use  for  all  problems  not  mentally  reducible  to  a  common 
denominator. 

Note.  —  The  example  used  in  the  above  illustration  could  be  more 
readily  solved  by  mentally  reducing  both  fractions  to  sixths,  when  the 
quotient  would  be  readily  seen  to  be  f.  The  method  of  inversion  is  only 
shorter  when  the  fractions  have  much  larger  terms,  or  are  not  readily 
reducible. 

A  fraction  may  be  divided  by  a  whole  number  by  dividing  the  numera- 


62  ELEMENTS  OF  BUSINESS  ARITHMETIC 

tor,  or  by  multiplying  the  denominator.  The  latter  divides  by  decreas- 
ing the  size  of  the  denominator.  A  whole  number  may  be  divided  by 
a  fraction  by  reducing  it  to  a  similar  fraction  and  dividing  the 
numerators. 


Divide: 

1.    T^byf.       3.   ^by^^^.     5. 

fibyxir. 

7.   tfbyf. 

9.  HbyH. 

2.   Ifbyf.      4.   ilbyf.       6. 

H  by  f  ?. 

8.   fby^. 

10.  ^byl6. 

11.   tfby^f.                      13. 

If  by  14. 

15. 

8bys. 

12.  xfirbyM.                     14. 

1  by  13. 

16. 

213  by  H. 

47.  Decimal  Fractions.  By  definition  (Sec.  38),  a  deci- 
mal fraction  is  a  common  fraction  whose  denominator  is  1 
with  ciphers  annexed.  Thus,  ^y^^,  loVW  ^^^  -^^  are 
decimal  fractions. 

These  decimal  fractions  may  be  written  without  the  de- 
nominator (Sec.  18),  by  pointing  off  from  the  right  of  the 
numerator  one  less  than  the  number  of  digits  in  the  denomi- 
nator. This  would  always  be  equal  to  the  number  of  ciphers 
in  the  denominator.  The  above  decimal  fractions  would 
thus  be  written  as  decimals :   .105,  .0017,  and  210.5. 

The  position  of  the  decimal  point  in  a  decimal  always  indi- 
cates, then,  the  number  of  ciphers  at  the  right  of  the  1  in 
the  denominator,  if  written  as  a  decimal  fraction.  Decimals 
may  thus  be  changed  to  common  fractions  by  writing  as 
decimal  fractions,  and  then  reducing,  if  desired,  to  lower 
terms. 

1.  Write  as  decimals :  ^^,  jiUh^  j^U^  f  |^,  and  m%. 

2.  Write  as  decimal  fractions  :  5.625,  .00052,  1.1,  and  32.0225. 

3.  Change  to  decimal  fractions  and  reduce  to  lower  terms :  .125,  .00875, 
.0625,  75.015,  and  300.05 

48.  Changing  Fractions  to  Decimals. 

(a)  When  the  denominator  of  the  required  fraction  is 
known. 


FRACTIONS  63 


Example.     -^  =  how  many  thousandths  ? 


A 


=  A  of  \m  =  -M^  thousandths  or  .ISTJ. 


Ordinary  business  problems  do  not  require  the  use  of  deci- 
mals beyond  the  thousandths.  Practice  in  reduction  from 
fractions  to  decimals  should  largely  be,  then,  to  acquire  facil- 
ity in  changing  to  tenths,  hundredths,  and  thousandths,  and 
in  writing  at  sight  the  decimal  form  of  the  simpler  fractions. 
After  the  method  of  reduction  is  understood,  these  decimal 
equivalents  should  be  drilled  upon  until  they  can  be  written 
at  sight. 

(5)  When  merely  a  change  of  form  is  desired. 

Example.     Reduce  -f^  to  a  decimal. 
16)3.0000 
.1875 
This  is  the  method  of  reduction  usually  given.     It  consists 
of  annexing  decimal  ciphers  to  the  numerator,  and  dividing 
by  the  denominator.     The  business  necessities  for  the  use 
of  this  method  are  not  frequent.      It  should,  however,  be 
thoroughly  understood. 


Change  to  Decimals: 

To  hundredths. 

To  thousandths. 

1.  /^                6.   1 

1.   jh 

6.   H 

2.   /^                7.   1 

2.   zh 

7.   ^ 

3.   M                8.   f 

3.   M 

8.  M 

4-   M                9-   A 

4.   fa 

9.   H 

5.   tW            10.    ^ 

5.   It 

10.   If 

Reduce  to  Decimals: 

Do  not  carry  beyond  five  places. 

1.  A  6.  i 

2.  li  7.   ^^ 

3.  ^  8.   il 

4.  Yh  9.   H       ■ 

5.  ^2  10.   3f 


64  ELEMENTS  OF  BUSINESS  ARITHMETIC 

MISCELLANEOUS  PROBLEMS  IN  FRACTIONS 

1.  An  estate  is  divided  among  three  sons  so  that  the  first  gets  ^,  the 
second  ^j^,  and  the  third  the  remainder,  $3600.  What  is  the  amount  of 
the  estate? 

2.  If  a  clerk  spends  |  of  his  weekly  salary  for  board,  ^  for  clothing, 
and  i  for  books  and  papers,  and  has  left  $  4,  what  is  his  salary  ? 

3.  If  I  sell  a  house  for  $3600,  thereby  gaining  |,  what  was  the  cost 
of  the  house  ? 

4.  A  real  estate  agent  rents  a  house  for  $850,  which  is  f  of  its  cost. 
What  is  its  cost  ? 

5.  A  book  dealer  paid  $21  for  a  set  of  books  and  sold  them  for  $24. 
The  gain  was  what  part  of  the  cost? 

6.  A  farmer  sold  two  horses  for  $48  each.  On  one  he  lost  f  of  the 
cost,  on  the  other  he  gained  j  of  the  cost.  How  much  did  he  gain  or 
lose  by  the  transaction  ? 

7.  If  3  yd.  of  cloth  cost  37^  ^,  how  much  change  would  you  receive 
from  a  $5  bill  if  you  buy  15  yd.  ? 

8.  A  grain  dealer  had  $14,000.  He  spent  f  of  it  for  wheat  at  75  ^  a 
bushel,  and  f  of  the  remainder  for  oats  at  30  j^^  a  bushel.  How  many 
bushels  of  each  did  he  buy,  and  how  much  money  had  he  left  ? 

9.  A  man  spent  i  of  his  money  for  a  suit,  |  of  the  remainder  for  a 
shotgun,  and  has  left  $40.     How  much  had  he  at  first? 

10.  A  locomotive  runs  |  of  a  mile  in  f  of  a  minute.  At  what  rate 
per  hour  does  it  run  ? 

11.  The  silver  dollar  weighs  412.5  grains ;  ^  of  its  weight  is  alloy. 
How  many  grains  of  pure  silver  are  there  in  one  dollar  ? 

12.  Gold  is  19.36  times  as  heavy  as  water;  copper,  8.97;  lead,  11.36. 
Find  the  weight  of  a  cubic  foot  of  each,  if  a  cubic  foot  of  water  weighs 
62^  lb. 

13.  The  total  crop  of  cotton  in  the  United  States  in  a  certain  year 
was  10,758,000  bales.  Of  this  amount  6,482,849  bales  were  exported  to 
Europe.  What  fraction  (decimal)  of  the  crop  was  exported  ?  Retained 
at  home? 

14.  If  a  coal  dealer  gains  ^  by  selling  coal  for  $8  a  ton,  how  much 
would  he  gain  on  a  sale  of  8.8  tons  ? 


FRACTIONS  65 

15.  After  paying  $  74.85  for  mileage,  ^  37.50  for  hotel  bills,  and  $  13.65 
for  sundry  expenses,  a  traveling  agent  finds  that  he  has  expended  |  of 
his  money.     How  much  had  he  at  first  ? 

1^.   pind-the  value  of  a  sheep  which  dressed  as  follows : 

Leg 22.51b.,  121;? 

Loin 17.5  lb.,  O^f 

Rib 14.8  lb.,  9f^ 

Chuck 19.6  1b.,  2f  J? 

17«-  The  price  of  corn  as  quoted  at  the  close  of  the  market  each  day 
was  as  follows:  Monday,  58^^;  Tuesday,  58|^;  Wednesday,  56^;''; 
Thursday,  J^7f^  ;  Friday,  57f  ^  ;  Saturday,  56|^.  Find  the  average  price 
for  the  week. 

18.  A  can  do  a  piece  of  work  in  4  days,  B  can  do  it  in  5  days.  How 
long  will  it  take  them  both  to  do  it  ? 

19.  A  general  store's  sales  of  dry  goods  for  a  month  amounted  to 
$  6300,  and  f  of  the  sales  of  dry  goods  was  f  the  sales  of  groceries.  What 
was  the  sales  of  groceries  ? 

20.  A  bank  teller  received  during  the  day  $  60,000  in  silver  and  paper 
money.  There  was  f  as  much  silver  as  paper  money.  How  much  of 
each  did  he  receive  ? 

21.  An  automobile  cost  $  1600.  If  f  the  cost  of  the  automobile  is  4 
times  the  cost  of  a  carriage,  what  is  the  cost  of  a  carriage? 

22.  A  grocer  bought  eggs  at  the  rate  of  4  for  5  f  and  6old  them  at  the 
rate  of  5  for  9  ^.     How  much  did  he  gain  on  each  dozen  ? 

23.  A  piece  of  cloth  is  20  yd.  long  and  |  yd.  wide.  How  wide  is 
another  piece  which  is  12  yd.  long  and  contains  as  many  square  yards 
as  the  first  ? 

24.  If  I  pay  $48  for  a  buggy  after  receiving  a  discount  of  |,  and  a 
further  discount  of  f  of  the  latter  price  for  cash,  what  was  the  asking 
price? 

25.  A  broker  sold  stocks  at  $  82  and  gained  :^o-  What  would  he  have 
gained  or  lost  had  he  sold  them  a  few  days  later  when  they  were  quoted 
at  $64? 

26.  If  an  ordinary  gas  burner  consumes  ^q  cu.  ft.  of  gas  per  second, 
what  would  be  the  cost  per  hour  to  light  a  room  with  50  burners  at  $1.25 
per  thousand  cubic  feet?  If  a  Welsbach  burner  consumes  ^  as  much  gas, 
how  much  would  be  saved  in  a  day  of  6  hours  by  installing  Welsbach 
burners  ?     How  much  in  a  month  of  30  days  V 


i 


66  ELEMENTS  OF  BUSINESS  ARITHMETIC 

27.  A  owned  f  of  a  store  and  sold  |  of  his  share  to  C.  C  sold 
I  of  what  he  bought  to  B  for  $4000.  At  this  rate,  what  was  the  store 
worth  ? 

28.  James  and  John  hire  a  pasture  for  $  35.  James  puts  in  4  cows 
and  John  puts  in  3.     What  must  each  pay? 

29.  A  merchant  sold  a  quantity  of  coffee  for  $1280,  and  thereby 
gained  ^  of  the  cost.  If  he  had  sold  it  for  $  1000,  would  he  have  gained 
or  lost,  and  how  much  ? 

30.  The  net  profits  of  a  business  for  two  years  were  $6400.  The 
second  year's  profits  were  |  greater  than  the  first  year's.  What  were  the 
profits  each  year  ? 

31.  I  paid  $22,500  for  two  farms.  If  f  of  the  cost  of  one  is  equal  to 
f  of  the  cost  of  the  other,  what  did  I  pay  for  each  one  ? 

32.  A  merchant  bought  300  crates  of  peaches  at  87^^  a  crate.  He 
sold  I  of  them  at  an  advance  of  10^  a  crate,  |  of  the  remainder  at  80 j^  a 
crate,  and  the  remainder  at  a  loss  of  3^  ^  a  crate.  What  did  the  mer- 
chant gain  or  lose? 

33.  A  man  invested  ^  of  his  money  in  bonds,  ^  of  it  in  real  estate, 
I  in  mining  stock,  and  the  balance,  $3900,  in  bank  stock.  How  much 
did  he  have  in  all  ?     How  much  in  each  investment  ? 

34.  A  man  spent  f  of  his  money  for  a  house,  invested  ^  of  the  re- 
mainder in  stocks,  and  had  $3200  left.     How  much  had  lie  at  first? 

35.  A  tree  fell,  breaking  in  three  pieces.  The  first  wa^  f  as  long  as 
the  second,  and  the  third  was  I  as  long  as  the  other  two  pieces.  What 
was  the  length  of  each  piece,  if  the  total  length  was  180  ft.  ? 

36.  A  speculator  invested  ^  of  his  money  and  $600  in  land,  ^  of  his 
money  and  $250  in  bank  stock,  I  of  his  money  and  $144  in  bonds,  and 
the  remainder,  which  was  $1400,  in  a  house  and  lot.  What  did  he  in- 
vest in  each  kind  of  property  ? 

37.  George's  money  is^'f/of  James'.  James'  money  is  |  of  Clara's. 
Clara's  is  1^  times  Daniel's.  How  much  had  each,  if  ^  of  George's 
money  is  $60? 


VI 

MEASURES  OF  LENGTH 

49.  Measurement.  Measurement  of  quantity  enters  so 
largely  into  life  that  to  make  arithmetic  really  practical,  our 
concepts  of  the  units  of  measurement  should  be  very  accu- 
rate. 

The  first  efforts  of  the  learner  should  be  toward  the 
building  of  accurate  concepts  of  the  various  units,  rather 
than  toward  proficiency  in  repeating  tables  or  in  changing 
to  higher  or  lower  units. 

Note.  —  Pupils  should  have  the  actual  measurement  units  present  to 
their  senses.  Do  not  make  the  mistake  of  talking  of  rods,  miles,  acres, 
etc.,  without  bringing  these  quantities  actually  before  you.  Use  a  tape 
or  string  a  rod  long ;  view  and  walk  a  mile,  or  view  and  walk  around 
an  acre,  etc.  Practice  judging  length,  extent,  or  weight,  and  test  your 
accuracy  by  measurement. 

50.  The  Unit  of  Length.  The  unit  for  measuring  length 
is  the  yard.  Formerly  the  unit  for  the  United  States  was 
the  same  as  the  English  yard,  but  the  law  of  1893  made  the 
yard  fffy  of  the  international  unit,  the  meter.  A  standard 
yard  is  kept  in  the  Bureau  of  Standards  at  Washington 
(Sec.  127). 

TABLE 

There  are:        12  inches  ('^)  in  1  foot  ('). 
3  feet  in  1  yard  (°). 
5J  yards  in  1  rod. 
320  rods  in  1  mile. 

1  mile  =  5280  feet  or  63,360  inches. 
67 


68  ELEMENTS  OF  BUSINESS  ARITHMETIC 

51.  Surveyor's  Measure.  The  unit  of  land  measure  has 
long  been  the  Gunter's  chain,  4  rods,  or  6Q  feet  long.  This 
was  divided  into  100  parts  or  links,  each  link  being  7.92 
inches  long.     Eighty  of  these  chains  make  a  mile. 

In  civil  engineering  and  at  the  customhouse,  the  inch  and 
foot  are  divided  into  tenths,  hundredths,  and  thousandths,  in 
lieu  of  the  usual  subdivisions  of  halves,  quarters,  eighths,  etc. 
This  is  indicative  of  a  tendency  in  all  measurements  toward 
a  larger  use  of  decimal  divisions.  Some  of  the  advantages 
of  a  decimal  system  of  measurement,  the  strongest  argument 
for  the  metric  system,  are  thereby  secured. 

52.  Nautical  Units.  Numerous  special  units  have  become 
established  by  usage  in  particular  vocations.  Many  of  these 
are  of  local  use,  others  have  varying  values  in  different 
localities,  and  still  others  have  become  or  are  becoming  ob- 
solete. A  few  nautical  units  of  length  or  distance  are  here 
given. 

There  are :     6  feet  in  1  fathom. 

1.15  statute  miles  in  1  geographic  (some- 
times called  nautical)  mile  or  knot. 

3  geographic  miles  in  1  league. 

60  geographic  miles  in  1  degree. 

•69.16  statute  miles  in  1  degree  of  the  earth's 
equator. 

360  degrees  in  1  circumference  of  the  earth. 

CLASS  EXERCISES 

1.  Mark  off  1  foot  in  length  on  the  board.     Scan  it  carefully. 

2.  Mark  off  1  foot  without  using  the  ruler.     Test  accuracy. 

3.  Draw  a  horizontal  line  i  foot  long ;  a  vertical  line ;  an  oblique 
line.     Test  them. 

4.  Estimate  the  number  of  feet  in  length  of  your  desk  top ;  its  width; 
the  length  and  width  of  the  door;  of  the  window.  Verify  your  es- 
timates by  measuring. 


MEASURES  OF  LENGTH  69 

5.  Estimate  the  height  of  the  room ;  its  width  ;  its  length.     Test. 

6.  Practice  judging  length  of  articles  in  the  room,  and  of  distances, 
until  you  can  estimate  a  foot  with  considerable  accuracy.  An  inch. 
A  yard. 

7.  Carefully  measure  off  a  rod.  Estimate  length  in  rods  until  you 
can  do  so  accurately. 

8.  Make  a  chain,  either  Gunter's  or  100-foot  chain.  In  company  with 
another  pui^il,  if  it  can  be  arranged  after  school  hours,  measure  off  a 
mile.  Mark  its  limit  by  some  signal,  so  that  the  eye  may  judge  the  dis- 
tance. Walk  the  mile.  Estimate  distances  in  miles  and,  if  possible, 
verify  estimates. 

PROBLEMS 

1.  How  many  inches  in  7  ft.?    In  3  yd. ?    In  a  rod? 

2.  How  many  feet  in  90  in.  ?    In  4  rd.  ?    In  a  half  mile  ? 

3.  A  ship  travels  18  knots  per  hour.  How  many  miles  does  she  travel 
in  6  hours  ? 

4.  A  garden  is  90  ft.  square.  How  many  yards  of  fence  will  it  take 
to  inclose  it? 

5.  A  well  is  25  yd.  deep.  What  will  be  the  cost  of  a  pump  stock, 
that  reaches  to  the  bottom,  at  8/^  per  foot? 

6.  If  potato  rows  are  3  ft.  apart,  how  many  rows  are  there  in  a  lot 
4  rd.  wide  ? 

7.  A  gardener  has  a  bed  16  ft.  long  and  6  ft.  wide.  He  wishes  to 
have  4  rows  of  plants,  6  in.  apart,  in  the  row.  How  many  plants  will  it 
take  ?     How  far  apart  are  the  rows  ? 

8.  How  many  miles  does  a  boy  ride  in  a  month  of  26  days,  if  he 
rides  342  rd.  to  and  from  work  each  day  ? 

9.  If  a  man  digs  a  ditch  3  rd.  long  in  a  day,  how  long  will  it  take 
him  at  the  same  rate  to  dig  a  ditch  I  mi.  long? 

10.  If  railroad  ties  are  laid  18  in.  apart  (from  center  to  center),  how 
many  ties  will  it  take  to  lay  a  mile  of  track?  What  will  they  cost  at 
75  f  apiece  ? 

11.  What  will  it  cost  to  place  a  hedge  around  a  lot,  the  distance  around 
which  is  72  rd.,  if  the  plants  are  placed  6  in.  apart  and  cost  |4.25  per 
hundred  ? 


70  ELEMENTS  OF  BUSINESS  ARITHMETIC 

12.  The  distance  between  two  cities  is  12  mi.  What  will  the  wire 
to  build  a  telephone  line  cost  at  «|2.90  per  hundredweight,  if  it  weighs 
l^lb.  to  the  rod? 

13.  A  barn  roof  is  84  ft.  long.  What  will  it  cost  to  place  an  eave- 
trough  along  two  sides,  if  the  trough  costs  6d^  per  10  ft.  length? 

14.  The  length  of  a  rectangular  lot  is  100  yd.,  its  width  40  yd. 
What  will  it  cost  to  fence  it  with  wire  netting  at  $3.60  per  150  ft.,  and 
posts,  set  10  ft.  apart,  at  35  j^  each  ? 

15.  A  water  company  wishes  to  lay  a  line  of  pipe  along  a  mile  of 
street.    If  the  pipe  is  worth  f  1.25  per  foot,  what  will  it  cost  ? 

16.  A  gentleman  has  a  field,  the  perimeter  of  which  (i.e.  the  distance 
around  it)  is  320  rd.  He  wishes  to  build  a  fence  of  eight  wires,  with 
posts  set  8  ft.  apart  around  it.  If  the  wire  weighs  1  lb.  to  the  rod  and 
costs  f  2.50  per  hundredweight,  and  the  posts  cost  17^  each,  what  will  it 
cost  for  material  to  build  the  fence? 

17.  What  will  it  cost  to  fence  a  lot,  with  boards  16  ft.  long  and  4  in. 
wide,  at  10)^  per  board,  if  the  perimeter  of  the  lot  is  63  rd.  and  the  fence 
is  to  be  6  boards  high? 

18.  My  lot  is  32  rd.  long  and  20  rd.  wide.  I  built  a  picket  fence 
around  it,  using  pickets  4  in.  wide,  and  placing  them  2  in.  apart.  How 
many  pickets  were  required?     What  did  they  cost  at  $3.25  per  M? 

19.  A  plot  of  ground  is  4  rd.  long  and  2  rd.  wide.  How  many  straw- 
berry plants  will  it  take  to  set  the  plot  in  rows  2  ft.  apart,  running 
lengthwise,  if  the  plants  are  set  10  in.  apart  in  the  row? 

20.  What  will  the  material  to  build  125  mi.  of  railroad  cost,  if  the 
rails  are  30  ft.  long,  weigh  25  lb.  to  the  foot,  and  cost  $28  per  ton  of 
2000  lb.,  and  the  ties,  costing  75^  apiece,  are  laid  3050  to  the  mile? 

21.  What  will  it  cost  to  build  an  electric  line  between  two  cities,  15 
mi.  apart;  the  rails  weighing  25  lb.  to  the  foot,  and  costing  $18  per  ton; 
the  ties  costing  70  ^  apiece,  laid  2  ft.  apart ;  the  wire  (one  strand)  weigh- 
ing 5  lb.  to  the  rod  and  costing  $10  per  hundredweight;  the  posts  being 
set  19  rd.  apart,  and  costing  80 j*  apiece? 

22.  A  tennis  court  has  four  lines  78  ft.  long,  two  lines  36  ft.  long, 
two  lines  27  ft.  long,  and  one  line  31  ft.  long.  What  will  the  tape  to  mark 
these  lines  cost  at  i^  per  foot? 

23.  How  many  yards  of  carpet  will  be  required  for  a  flight  of  15  steps 
1  ft.  wide  and  6  in.  high,  and  a  landing  6  ft.  8  in.  wide?  What  will 
it  cost  at  70  ^  per  yard  ? 


MEASURES  OF  LENGTH  71 

24.  The  two  aisles  of  a  church  are  each  6  ft.  wide  and  85  ft.  long. 
At  $  1.85  per  linear  yard,  what  will  the  carpet  cost  to  cover  them? 

25.  In  surveying  the  route  for  a  proposed  railroad  the  surveyors  ap- 
plied the  Gunter's  chain  6450  times.  How  many  miles  of  road  ?  How 
many  rails,  30  ft.  long,  would  it  require  ?  How  many  ties,  counting 
3050  to  the  mile? 

26.  A  boat  is  rowed  at  the  rate  of  10  knots  an  hour.  The  current 
runs  at  the  rate  of  4  knots  an  hour.  If  the  boat  is  rowed  with  the  cur- 
rent, how  many  miles  will  it  go  in  10  hr.  ? 

27.  The  elevator  in  the  Washington  monument  goes  500  ft.  above 
the  base.  How  many  rods  of  cable  in  the  eight  strands  extending  from 
bottom  to  top  ? 


VII 

MEASURES  OF   AREA 

53.  Area  Units.  Area  is  the  extent  of  surface.  The  units 
for  its  measurement  are  rectangular  in  form  and  correspond 
to  the  units  of  length.  They  are  the  square  inch,  square 
foot,  square  yard,  square  rod,  and  square  mile.  The  stand- 
ard unit  is  the  square  yard.  For  ordinary  surfaces,  the 
square  inch  and  square  foot  are  most  frequently  used.  The 
square  mile  is  used  only  in  land  measure,  together  with  a 
special  unit,  the  acre. 

Table 
There  are : 

144  square  inches  (sq.  in.)  in  1  square  foot  (sq.  ft.) 

9  square  feet  in  1  square  yard  (sq.  yd.) 

30^  square  yards  in  1  square  rod    (sq.  rd.) 

160  square  rods  in  1  acre  (^0 

640  acres  in  1  square  mile  (sq.  mi.) 

36  square  miles  in  1  township       (Tp.) 

CLASS  EXERCISES 

1.  Study  the  accompanying  figure  of  a  square  inch.  Cut  from  paper 
a  square  inch  without  using  a  ruler.  Test  accuracy.  Try  it  until  nearly 
correct. 

2.  Draw  a  square  inch.     Verify.     Practice. 

3.  Estimate  the  number  of  square  inches  in  one  side  of  your  book 
cover.  Test  accuracy.  Estimate  the  area  in  square  inches  of  pages  in 
different  books,  of  desk  top,  of  window  pane,  of  pictures,  and  of  fresh 
air  or  foul  air  register.     Test  the  accuracy  of  each  estimate. 

72 


MEASURES  OF  AREA 


73 


4.  Practice  estimating  areas  in  square  inches  until  you  acquire 
considerable  accuracy.  Use  the  ruler  only  after  the  area  has  been 
judged. 

5.  Develop  power  of  estimating  area  in  square  feet  in  the  same  way 
as  above.     Measure,  cut,  draw,  and  practice  estimating. 

6.  Measure  off  a  square  yard.  Practice  estimating  in  square  yards 
until  a  fairly  clear  concept  is  in  your  mind.  Less  time  and  practice  are 
necessary  for  this  than  for  the  square  inch  and  foot,  because  less  fre- 
quently needed. 

7.  In  the  same  way  develop  concept  of  a  square  rod.  Practice  for 
accuracy. 

8.  The  pupils  of  a  class  should  measure  off  an  acre,  using  chains  of 
their  own  making.  Let  them  mark  the  corners  with  flags  and  walk 
over  the  ground  in  an  effort  to  fix  the  size  of  an  acre.  Then  have  pupils 
estimate  in  acres  and  verify  their  estimates  by  measurement. 

9.  If  possible,  view  a  square  mile  or  section  of  land.  Use  what  oppor- 
tunities you  have  for  fixing  its  size  in  your  mind. 

54.  Area  of  Rectangles.  Finding  the  area  of  anything  is 
merely  finding  the  number  of  square  units  on  its  surface. 
In  square-cornered  surfaces,  or  rectangles,  these  units  are 
always  in  rows.     In  the  accompanying  figure,  2  by  4  inches, 


A  Rectangle 


74  ELEMENTS  OF  BUSINESS  ARITHMETIC 

there  are  four  square  inches  along  one  edge  or  in  one  row, 
and  two  such  rows.  There  are,  therefore,  2  times  4  square 
inches  or  8  square  inches  in  the  figure. 

In  all  rectangles  there  are  as  many  square  units  in  one 
row  or  along  one  edge  as  linear  units  along  that  edge.  Like- 
wise, there  are  as  many  rows  of  square  units  as  linear  units 
along  the  other  edge.  To  find  the  area,  then,  the  number  of 
square  units  in  one  row  should  he  multiplied  hy  the  number  of 
rows. 

In  a  room  10  by  12  feet,  there  are  10  rows  of  12  square 
feet  or  120  square  feet  in  the  floor.  The  multiplicand  is 
thus  to  be  always  a  number  of  square  units,  and  the  product, 
therefore,  square  units. 

PROBLEMS 
Find  the  area  of  the  following  rectangles : 

1.  6  by  9  inches.  6.   35  by  46  feet. 
9  sq.  in.  X  6  =  54  sq.  in.  7.   15  by  36  rods. 

2.  13  by  24  rods.  8.   45  by  86  yards. 

3.  16  by  87  yards.  9.   32  by  54  inches. 

4.  41  by  65  feet.  10.   40  by  60  yards. 

5.  11  by  47  inches.  11.  24  by  48  feet. 

12.  A  barn  floor  is  40  by  60  ft.  What  will  it  cost  to  cement  it  at  12/ 
per  square  foot  ? 

13.  How  many  rolls  of  weather  paper  each  containing  100  sq.  ft. 
will  it  take  to  cover  a  roof  42  by  50  ft.  ? 

14.  A  board  fence  7  ft.  high  surrounds  a  lot  32  by  120  ft.  How 
many  square  feet  of  boards  in  this  fence?  What  would  it  cost  to  paint 
it  at  4  /  a  square  foot  ? 

15.  A  box  is  6  by  8  in.  and  4  in.  high.  How  many  square  inches  in 
its  4  sides? 

Note.  —  Think  of  the  sides  as  a  rectangle  having  4  rows  of  28  sq.  in. 

16.  A  room  is  12  by  18  ft.  and  10  ft.  high.  How  many  square  feet  in 
the  four  walls?    lu  the  floor  and  ceiling? 


MEASURES  OF  AREA  75 

17.  What  will  it  cost  to  tint  the  four  walls  of  a  room  15  by  18  ft.  and 
10  ft.  high,  at  40  ;*  a  square  yard  ?  How  many  square  yards  in  the  ceil- 
ing? What  will  be  the  cost  of  tinting  the  ceiling  at  the  same  rate? 
What  was  the  total  cost  of  plastering  the  room  at  25^  per  square  yard? 

18.  What  will  it  cost  to  build  a  walk  of  stone,  6  by  360  ft.,  at  60  ^ 
per  square  foot  ? 

19.  A  barn  is  40  by  80  ft.  and  25  ft.  high.  What  will  it  cost  to  paint 
the  four  sides  at  25 f  a  square  yard? 

20.  A  hall  floor  is  15  by  30  ft.  How  many  tiles  6  by  6  in.  will  it 
take  to  lay  the  floor  ? 

21.  There  are  48  sq.  ft.  of  boards  on  the  side  of  a  building.  If  each 
board  is  6  in.  wide,  and  there  are  12  of  them,  how  long  are  they? 

22.  What  will  it  cost  to  calcimine  the  walls  and  ceiling  of  a  room 
18  by  15  ft.,  and  9  ft.  high,  at  20^  per  square  yard? 

23.  A  schoolroom  is  30  by  40  ft.  What  will  it  cost  to  put  a  slate 
board  along  one  side  and  end,  the  slate  being  4  ft.  wide  and  costing  18;^ 
per  square  foot? 

24.  How  many  bricks  4  by  8  in.  will  be  needed  for  a  walk  36  yd. 
long  and  4  yd.  wide?    What  will  they  cost  at  $6.50  per  M? 

25.  How  many  square  feet  of  sidewalk  6  ft.  wide  will  be  required  to 
surround  a  lot  250  by  360  ft.?  What  will  it  cost  to  lay  it  at  12;^  a 
square  foot? 

26.  The  owner  of  a  lot  of  ground  1080  ft.  long  and  600  ft.  wide  cuts 
two  streets,  each  75  ft.  wide,  through  the  middle,  one  running  east  and 
west,  and  the  other  running  north  and  south.  How  many  acres  has  he 
left?     What  will  brick  6  by  9  in.  cost  to  pave  these  streets  at  $  7  per  M? 

27.  A  farmer  owned  a  rectangular  piece  of  land  40  by  80  rd.  He 
sold  four  lots,  7  by  10  rd.,  8  by  25  rd.,  10  by  15  rd.,  and  15  by  20  rd.  How 
many  acres  did  he  sell?  How  many  remained?  How  many  rods  of 
fence  will  it  take  to  fence  the  remaining  land,  if  the  lots  are  taken,  one 
from  each  corner? 

55.  Squaring  a  Number.  A  rectangle  which  contains  the 
same  number  of  rows  of  square  units  as  there  are  square 
units  in  each  row  is  called  a  square.  The  number  of  units  in 
each  row  of  a  square  is  called  the  square  root. 


76  ELEMENTS  OF  BUSINESS  ARITHMETIC 

CLASS  EXERCISES 

1.  The  square  on  3  ft.  has  how  many  rows  and  how  many  square 
units  in  each  row?     How  many  square  units  in  the  square,  or  its  area? 

2.  A  square  on  9  in.  has  how  many  square  inches  in  each  row? 

3.  When  a  square  contains  2.5  sq.  in.,  it  is  how  many  inches  long  and 
wide? 

4.  What  is  the  square  root  of  9  sq.  ft.  ?    Of  16  sq.  ft.  ?    Of  25  sq.  ft.  ? 

5.  What  is  the  square  root  of :  81,  49,  100,  225,  2500,  625,  169? 

6.  Drill  upon  the  following  table  : 

12  =  1  62  =  36  112  =  121  162  ^  256 

22  =  4  72  =  49  122  ^  144  202  =  400 

32  =  9  82  =  64  132  =  169  252  =  625 

42  =  16  92  =  81  142  =  196  502  ^  2500 

52  =  25  102  =  100  152  =  225  1002  =  10000 

FINDING  THE  SQUARE  ROOT 

66.    Root  Periods  in  the  Square.     In  squaring     ][2  _  ^ 
numbers  from  1  to  10,  it  may  be  noted  that  the     2^  =  4 
square  of  a  number  composed  of  one  figure  never     32  —  9 
contains  less  than  one  or  more  than  two  figures.      ^2  _  iq 
Squaring   10   and   99,  it  will   be   seen  that   the     ^2  __  25 
square  of  a  number  composed  of  two  figures  has     ^2  _,  3^ 
not  less  than  three  nor  more  than  four  figures.      ^2  =  49 
In  the  same  way  it  may  be  seen  that  the  square     g2  __  54 
of  a  number  composed  of  three  figures  contains     92  _  g;[ 
not  less  than  five  nor  more  than  six  figures.     It  ■^q2  _  iqO 
may  be  concluded,  then,  that  for  every  two  fig-   992  _  g^Qi 
ures  in  a  square  there  is  one  figure  in  the  root. 
If  wie  mark  off  the  number  whose  square  root  we  are  seek- 
ing, into  periods  of  two  figures  each,  beginning  at  the  right, 
we  will  get  one  figure  in  the  root  for  each  period. 

57.  Square  of  a  Number  composed  of  Two  Figures.  Take 
a  paper  24  in.  square,  and  carefully  mark  it  off  into  square 
inches.     From  a  lower  corner  cut  out  the  square  on  20  in.. 


MEASURES  OF  AREA 


77 


and  from  the  opposite  upper  corner  cut  out  the  square  on 
4  in.  What  are  the  dimensions  of  the  two  oblongs  remain- 
ing after  the  squares  have  been  removed  ?  Observe  that  the 
square  on  20  in.  is  the  square  of  the  tens  of  the  number 
squared  (24),  and  that  the  other  square  is  the  square  of  the 
units  (4).  The  dimensions  of  the  two  oblongs  are  also, 
respectively,  the  same  as  the  tens  and  units  of  the  original 
number. 

The  area  of  the  square  of  a  number  composed  of  two  figures 
is,  then,  equal  to  the  square  of  its  tens  plus  two  rectangles  with 
the  tens  and  U7iits  as  dimensions,  plus  the  square  of  its  units. 


A  Square 


f 


78 


ELEMENTS  OF  BUSINESS  ARITHMETIC 


r    f 

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whose  area  is  576  sq.  ft.,  or  extract  the  square  root  of  576. 

5'76  sq.  ft.  (24 
The    square  on  2  tens,  or 
20  sq.  ft.  X  2  =  40  sq.  ft. 


4  00 


_4  sq.  ft. 
44  sq.  ft. 
Sq.  ft.  in  one  row  of  2  rec- 
tangles and  smaller  square. 


1  76  sq.  ft.,  the  area  of  2 
rectangles  and  smaller 
square. 

1  76  sq.  ft.  the  square  feet 
in  4  rows. 


MEASURES  OF  AREA  79 

Marking  off  576  according  to  Sec.  56,  we  find  that  its 
root  will  contain  two  figures,  tens  and  units.  From  Sec.  57, 
also,  we  see  that  the  second  period,  or  6  hundreds,  contains 
the  square  of  the  second  or  tens  figure. 

Taking  out  the  square  of  the  largest  tens  (2  tens)  con- 
tained in  6  hundreds,  or  4  hundreds,  we  have  left  176  sq.  ft. 
as  the  area  of  the  remaining  two  rectangles  and  the  smaller 
square.  We  know  (Sec.  57)  that  the  rectangles  have  each 
two  tens,  or  20  sq.  ft.,  along  one  edge  or  in  one  row.  We  do 
not  as  yet  know  the  dimensions  of  the  smaller  square,  but 
we  know  they  are  the  same  as  the  other  dimension  of  the 
rectangles. 

If  the  area  of  the  rectangles  is  nearly  176  sq.  ft.,  and  the 
number  of  square  feet  along  one  edge  of  both  is  40 ;  then 
there  must  be  nearly  as  many  rows  of  40  sq.  ft.  as  40  sq.  ft. 
is  contained  in  176  sq.  ft.,  or  4  rows.  If  this  is  correct,  the 
smaller  square  is  4  ft.  square,  and  along  one  edge  there 
would  be  4  sq.  ft. 

Along  one  edge  of  both  the  rectangles  and  the  smaller 
square,  there  would  be,  then,  44  sq.  ft.,  and  if  4  rows  wide, 
the  area  of  the  rectangles  and  square  would  be  176  sq.  ft., 
or  the  same  as  the  area  remaining  after  removing  the  larger 
square. 

The  other  dimension  of  the  rectangles,  therefore,  must  be 
4  ft.,  and  the  units  figure  4,  making  the  square  root  24. 

STATEMENT  OF  PROCESS 

The  principles  applied  in  finding  the  square  root  of  larger 
numbers  are  the  same  as  when  the  root  contains  two  figures. 
The  two  periods  to  the  left  are  treated  as  containing  the 
square  of  the  number  composed  of  two  figures,  and  when 
these  are  found,  they  are  considered  as  the  known  tens'  figure, 
and  the  units'  figure  is  sought  in  the  next  period  to  the  right, 
and  so  on,  until  the  complete  root  has  been  found. 


80  ELEMENTS  OF  BUSINESS  ARITHMETIC 

The  following  is.  a  summary  of  the  steps  in  the  method  : 

(1)  Beginning  at  the  right,  mark  off  periods  of  two  figures 
each. 

(2)  By  inspection  use  the  square  root  of  the  largest  square 
contained  in  the  left-hand  period  as  the  first  figure  of  the 
root,  and  subtract  its  square  from  that  period,  bringing 
down  the  next  period  to  the  right. 

(3)  Treating  the  first  figure  thus  obtained  as  the  tens  and 
adding  a  cipher,  we  have  the  number  of  squares  along  one 
edge  of  one  rectangle.  Multiplying  this  by  2  and  dividing 
the  product  into  the  number  brought  down  (allowing  some 
for  the  area  of  the  smaller  square),  the  probable  width  of 
the  rectangles  or  second  root  is  obtained. 

(4)  Add  the  root  to  the  number  of  squares  along  one 
edge  of  both  rectangles,  and  multiply  the  sum  by  the  same 
figure  (as  indicating  the  number  of  rows)  to  complete  the 
square. 

(5)  Subtract  this  from  the  number  brought  down,  and 
bring  down  the  next  period,  proceeding  as  in  (3)  and  (4). 

59.  Square  Root  of  Decimals  and  Fractions.  When  the 
square  consists  of  a  decimal  or  a  whole  number  and  decimal, 
mark  off  periods  right  and  left  from  the  decimal  point,  and 
proceed  as  in  whole  numbers.  The  root  figure  for  the  first 
decimal  period  would  be  tenths;  the  second,  hundredths, 
etc. 

The  square  root  of  a  fraction  may  be  found  by  extracting 
the  square  root  of  the  numerator  for  a  new  numerator  and 
of  the  denominator  for  a  new  denominator. 

60.  Applications  of  Square  Root  to  the  Right   Triangle. 

Draw  a  right  triangle  (Sec.  68)  having  a  base  of  3  inches 
and  a  perpendicular  of  4  inches.  What  is  the  length  of 
the  hypotenuse  ? 


MEASURES  OF  AREA 


81 


Erect  a  square  on  the  perpendicular.  Erect  a  square  on 
the  hypotenuse,  and  on  the  base.  How  many  square  inches 
in  the  squares  on  the  base  and 
on  the  perpendicular?  In  the 
square  on  the  hypotenuse  ? 

Draw  a  right  triangle  with  base 
and  perpendicular,  respectively, 
6  and  8  inches.  Draw  squares 
on  the  three  sides.  Compare  the 
area  of  the  square  on  the  hy- 
potenuse of  the  triangle  with 
those    on    the    other    two    sides. 

These  illustrate  the  geometrical  truth,  that  the  square  on  the 
hypotenuse  of  a  right-angled  triangle  is  equal  to  the  sum  of 
the  squares  on  the  other  two  sides. 


?I> 

\D    THE 

Square  Root  of: 

1. 

4225 

4.   53,361 

7. 

72,984 

10. 

^m 

2. 

5625 

5.   17,424 

8. 

1900.96 

11. 

\m 

3. 

1225 

6.  97,344 

9. 

97.8121 

12. 

■i^ 

13.  The  base  of  a  triangle  is  8  ft.,  the  altitude  6  ft.  What  is  the 
hypotenuse  ? 

14.  Find  the  base  of  a  triangle  whose  altitude  is  12  ft.,  hypotenuse 
16  ft. 

15.  The  hypotenuse  of  a  triangle  is  45  ft.,  the  base  27  ft.     Find  the 

altitude. 

16.  A  square  farm  contains  160  A.     What  is  the  length  of  one  side  ? 

17.  A  ladder  is  25  ft.  long  and,  when  the  foot  is  placed  15  ft.  from 
the  foot  of  a  wall,  just  reaches  the  top.     How  high  is  the  wall  ? 

18.  A  lot  is  in  the  form  of  a  right-angled  triangle,  whose  base  is  16 
chains  and  altitude  12  chains.  How  many  rods  of  fence  will  be  needed 
to  inclose  it  ? 


82  ELEMENTS  OF  BUSINESS  ARITHMETIC 

19.  Find  in  rods  the  diagonal  of  a  square  field  that  contains  12  acres. 

20.  What  is  the  distance  from  one  lower  corner  to  the  opposite  upper 
corner  of  a  room,  that  is  36  by  72  ft.  and  12  ft.  high  ? 

PRACTICAL  APPLICATIONS  OF  AREA 

61.  Pitch  of  Roofs  and  Roofing.  The  degree  of  slant  given 
to  the  sides  of  a  roof  is  called  its  pitch. 

When  the  height  of  the  gable  is  one  fourth  the  width  of 
the  building,  the  roof  is  said  to  have  one-fourth  pitch.  When 
the  height  is  one  half  the  width,  the  roof  has  one-half  pitch; 
when  five  eighths,  it  has  a  five-eighths  or  Gothic  pitchy  etc. 
One  half  is  the  pitch  commonly  used. 

In  estimating  the  cost  of  roofing  in  accordance  with  plans 
of  a  building  under  consideration,  it  is  sometimes  necessary, 
from  the  size  of  the  building  and  the  roof  pitch,  to  find  the 
dimensions  of  the  two  sides  of  a  roof. 

Knowing  the  width  of  the  building,  the  degree  of  pitch 
will  give  the  height  of  the  gable  or  the  perpendicular  of 
the  right  triangle,  one  half  the  width  of  building  forming 
the  base.  Applying  the  principle  of  Sec.  60,  the  length  of  the 
hypotenuse  or  the  width  of  one  side  of  the  roof  is  then  found. 
The  distance  the  eaves  extend  over  the  sides  of  the  building 
should  be  added  to  the  roof  width,  and  the  extension  of  the 
gables  over  the  ends  should  be  added  to  the  length  of  the 
building  for  the  roof  length. 

The  standard  width  of  a  shingle  is  4  inches.  This  may 
vary,  however,  according  to  grade  or  style.  In  ordinary 
shingles  there  is  no  uniformity  in  width,  but  the  average 
width  in  a  bundle  is  supposed  to  be  4  inches. 

The  standard  exposure  to  the  weather  is,  likewise,  4  inches. 
That  is,  in  laying,  all  but  4  inches  of  the  length  of  a  shingle 
is  covered.  Thus,  the  average  area  of  the  exposed  surface 
of  a  shingle  is  16  sq.  in. 

In  selecting  the  proper  widths  of  shingles  so  that  all  joints 


MEASURES  OF  AREA 

PITCH  OF  ROOFS 


83 


40' 
One-fourth  Pitch 


40' 
One-half  Pitch 


40' 
Five-eighths  or  Gothic  Pitch 


84  ELEMENTS  OF  BUSINESS  ARITHMETIC 

will  be  covered,  and  discarding  defective  shingles,  there 
is  more  or  less  waste,  varying  with  the  grade  of  shingle 
used.  One  ninth  is  the  amount  usually  allowed  for  such 
waste. 

Thus,  9  shingles,  at  16  sq.  in.  each,  would  cover  a  surface 
of  144  sq.  in.  or  1  sq.  ft.  With  i  allowance  of  waste,  10 
shingles  would  be  required  for  1  sq.  ft.,  and  1000  shingles 
for  100  sq.  ft.,  the  surface  unit.  It  is  called  the  square  (100 
sq.  ft.),  and  is  used  in  roofing,  flooring,  slating,  etc.  As 
bundles  are  usually  made  up  to  contain  250  shingles,  it 
would  take/oi^r  to  cover  a  square^  or  one  bundle  would  cover 
a  surface  of  25  sq.  ft.  Thus,  a  roof  24'  by  30'  would  be 
720  sq.  ft.  on  each  side,  or  1440  sq.  ft.  on  both  sides.  This 
would  make  14.40  squares  and  would  require  14 J  thousand 
shingles  or  58  bundles. 

The  outer  edge  of  the  roof  is  usually  laid  double.  This  is 
offset,  however,  by  the  width  of  the  ridge  board. 

Bundles  of  shingles  are  not  usually  broken,  so  if  the  sur- 
face shows  that  a  fraction  of  a  bundle  is  needed,  a  whole 
bundle  must  be  purchased. 

When,  for  any  purpose,  an  exposure  other  than  the  stand- 
ard 4-inch  is  allowed,  the  number  of  shingles  or  bundles  are 
found  in  the  usual  way,  and  the  result  is  modified  to  meet 
the  conditions.  Thus,  if  the  exposure  is  to  be  3''  instead  of 
4'',  \  more  shingles  will  be  required  ;  if  2'\  twice  as  many  ;  if 
5",  \  less;  and  if  6'',  ^  less.  If  the  width  of  the  shingle 
varies  from  the  standard  4-inch,  a  correction  may  be  made 
in  the  same  way. 

Roofing,  other  than  shingling,  is  also  figured  by  the  square. 
In  slating,  the  size  of  the  slates  varies  from  &'  x  12''  to  16" 
X  24".  The  number  of  slates  needed  per  square  would, 
therefore,  vary  from  533  to  86.  Contractors  usually  figure 
from  prepared  tables,  showing  the  number  of  slate  at  a  given 
size  per  square. 


MEASURES  OF  AREA 


85 


62.  Flooring  in  Wood.  Boards  for  flooring  are  tongued 
and  grooved.  This  entails  a  loss  in  width  of  |  of  an  inch  for 
each  board.  That  is,  a  board  bought  as  a  3-inch  board  will 
cover  but  2|-  inches  in  width.  Three-eighth  inch  loss  on 
3  inches  is  |-inch  loss  on  1  inch.  In  other  words,  each  1  inch 
of  board  width  purchased  will  cover  but  J  inches  of  floor 
space.  It  will  require  |  more  lumber,  therefore,  than  it 
would  if  there  were  no  waste. 


Tongued  and  Grooved 


How  much  flooring  would  be  required  for  a  room  24^  x  32' 
if  3"  flooring  were  used  ? 

24  sq.  ft.  X  32  =  768  sq.  ft.     Surface  area. 

768  sq.  ft.  + 1  of  itself  =  878  sq.  ft.     Flooring  required. 

Spruce  or  pine  flooring  is  made  3  in.,  4  in.,  or  sometimes 
51  in.  wide.  Hardwood  flooring  is  but  2  in.  or  2|  in.  wide, 
or  even  less.  In  the  same  way,  allowances  for  loss  from 
grooving,  for  the  usual  widths  of  flooring,  i^  as  follows  :  for 
2  in.  flooring,  add  -^^ ;  for  2^  in.,  add  -^y  ;  for  4  in.,  add  ^; 


86  ELEMENTS  OF  BUSINESS  ARITHMETIC 

and  for  5 J  in.,  add  -^^  of  the  actual  floor  space.  Prove  the 
correctness  of  these  fractions,  as  above.  Carpenters  charge 
by  the  square  for  laying  floors. 

PROBLEMS 

1.  The  roof  of  a  shed  is  12  by  20  feet.  How  many  slates  6  by  12 
inches  will  be  required  to  cover  it?  How  many  shingles?  How  many 
bundles  of  shingles  ? 

2.  What  will  it  cost  to  shingle  the  two  sides  of  a  roof,  if  each  side  is 
20  by  80  feet,  with  shingles  costing  $3.20  per  M? 

3.  A  barn  is  40  by  80  feet,  and  the  roof  is  J  pitch.  How  many 
shingles  must  I  buy  to  roof  it,  if  the  roof  extends  18  inches  over  each 
side  and  end? 

4.  The  roof  of  a  church  is  Gothic  pitch.  The  building  is  40  by  72 
feet,  and  the  roof  extends  2  feet  over  each  side  and  end.  If  shingles  are 
laid  3  inches  to  the  weather,  what  will  be  their  cost  at  $4.50  per  M? 

5.  The  roof  of  a  house  is  56  feet  long  and  each  side  is  18  feet  wide. 
There  are  two  verandas,  each  having  a  roof  10  by  20  feet.  What  will 
the  shingles  cost  at  $4.75  per  M,  if  they  are  laid  5  inches  to  the 
weather  ? 

6.  A  mill  is  120  feet  long.  If  each  side  of  the  roof  is  32  feet  wide, 
and  it  extends  2|  feet  over  each  side  and  end,  what  will  the  shingles  cost 
at  $  3.80  per  M,  if  they  are  laid  4  inches  to  the  weather  ? 

7.  Figure  the  flooring  lumber  bill  for  house  shown  on  pages  90-91,  as 
follows :  Hard  pine  at  $48  per  M  on  veranda,  living  room,  dining  room, 
and  entire  upper  floor,  except  bathroom,  which  is  to  be  tiled ;  two-inch 
maple  at  $72  on  kitchen;  two-inch  oak  at  $80  on  reception  hall,  library, 
and  parlor;  four-inch  soft  pine  on  porch,  at  $60;  spruce,  5^  inches,  at 
$34  on  laundry  and  cellar  in  basement. 

63.  Carpeting.  The  standard  width  of  Kidderminster  and 
Ingrain  carpets  is  36  inches.  That  of  Brussels,  Wiltons, 
Axminsters,  etc.,  is  27  inches.  Borders  are  usually  221 
inches  wide. 

Rugs,  whether  pattern  rugs  or  made  up,  are  sold  by  the 
piece.  All  other  carpeting  is  sold  from  the  roll  and  by 
linear  measure. 


MEASURES  OF  AREA  87 

The  direction  the  carpet  is  to  be  laid  will  often  determine 
the  cost,  as  there  might  be  more  waste  if  laid  one  way  than 
if  laid  the  other. 

The  length  of  each  strip  would  be  the  length  of  the  room 
plus  the  allowance  for  matching.  The  number  of  strips 
would  be  determined  by  dividing  the  width  of  the  room  by 
the  width  of  the  carpet,  counting  a  full  strip  for  all  fractions, 
as  any  extra  width  must  be  cut  off  or  turned  under.  Thus, 
for  a  room  15  by  24  feet,  the  strips  would  be  8  yards  long, 
and  with  an  allowance  of  9  inches  for  matching,  8^  yards 
long.  If  the  carpet  were  Brussels,  there  would  be  6  strips 
and  18  inches  additional,  requiring  7  strips.  7  strips  of  8J 
yards  each  would  be  57J  yards. 

PROBLEMS 

1.  How  many  yards  of  carpet  1  yard  wide  will  it  take  to  cover  a  floor 
15  by  18  feet? 

2.  A  room  is  18  by  20  feet.  How  many  yards  of  carpet  f  yard  wide 
will  it  take  to  cover  the  floor  ? 

3.  How  many  yards  of  carpet  27  inches  wide  will  be  required  to  cover 
the  dining  room  in  the  floor  plan  on  p.  90,  if  the  strips  run  length- 
wise ?    What  will  it  cost  at  ^  1.20  per  yard  V 

4.  If  the  library  shown  on  p.  90  is  carpeted  in  the  most  economical 
way,  with  carpet  30  inches  wide,  what  will  be  the  cost  at  $  1.25  per  yard? 

5.  What  will  it  cost  to  cover  the  kitchen  of  the  house  shown  on  p. 
90  with  linoleum  costing  $1.35  per  square  yard? 

6.  The  three  bedrooms  in  the  plan  of  the  second  floor  on  page  91  are 
to  be  covered  with  carpet  1  yard  wide,  8  inches  waste  in  matching. 
What  will  be  the  cost  at  95  ^  per  yard,  if  the  strips  run  in  the  direction 
to  leave  the  least  waste  ? 

7.  The  stairway  in  the  floor  plan  (pp.  90-91)  is  11  feet  4  inches  high, 
the  tread  of  each  stair  is  12  inches,  and  the  riser  8  inches.  What  will  be 
the  cost  of  carpet  at  $1.35  per  yard? 

8.  A  rug  8'  X  13'  is  placed  in  the  living  room  at  a  cost  of  $42.50. 
The  remaining  space  is  painted  at  a  cost  of  36  ^^^  a  square  yard.  What 
is  the  total  cost  for  covering  the  floor  ? 


88  ELEMENTS  OF  BUSINESS  ARITHMETIC 

64.  Lathing  and  Plastering.  The  square  yard  is  the  unit 
by  which  estimates  are  made  for  the  cost  of  lathing  and  plas- 
tering. Contracts  are  made  at  a  given  price  per  square  yard 
of  plastering,  or  lathing,  or  both. 

In  estimating  the  cost  of  lathing  and  plastering  for  scratch 
(first)  and  brown  (second)  coat  on  wood  lath,  the  follow- 
ing quantities  are  generally  allowed  for  100  square  yards  : 
1400  to  1500  laths  (laths  are  put  up  in  bundles  of  50,  and 
sold  by  the  bundle  or  thousand);  10  pounds  of  3-penny 
lathing  nails ;  2^  barrels,  or  500  pounds,  of  lime  ;  45  cubic 
feet,  If  loads,  or  15  barrels,  of  sand ;  and  4  bushels  of  hair 
(hair  is  put  up  in  bushel  bundles,  containing  5  packages). 
For  the  best  quality  of  white  coat,  the  estimate  is  90  pounds 
of  lime  to  50  pounds  of  plaster  of  Paris,  and  50  pounds  of 
marble  dust.  Stucco  is  put  up  in  bags  containing  100 
pounds  each,  and  is  sold  by  the  ton ;  900  to  1000  pounds  are 
required  for  100  square  yards  of  surface. 

Custom  varies  as  to  an  allowance  for  doors  and  windows. 
Some  contractors  make  no  allowance,  holding  that  the  extra 
care  and  time  needed  in  lathing  and  plastering  around  an 
opening  make  up  for  saving  of  material.  When  an  allow- 
ance is  made,  it  is  usually  20  sq.  ft.  for  each  opening,  and 
not  the  exact  measurement.  In  estimating  material  alone, 
allowance  is  usually  made  for  all  openings.  A  fraction  of  a 
square  yard  is  counted  as  a  square  yard  in  the  final  result. 

PROBLEMS 

1.  What  will  it  cost  to  plaster  a  room  20  x  24  ft.  and  10  ft.  high, 
at  20  ^  per  square  yard  ?  How  much  lime,  hair,  and  sand  will  it  take 
for  two  coats  ? 

2.  How  many  square  yards  in  the  walls  and  ceiling  of  a  room  18  by 
20  ft.  and  9  ft.  high  ?  How  many  laths,  and  how  much  stucco  will  it 
take  to  cover  it  ? 

3.  A  hall  is  5  X  15  ft.  and  10  ft.  high.  What  will  it  cost  to  lath  and 
plaster  it  at  25^  a  square  yard  ?    How  many  laths  will  it  take  ? 


MEASURES  OF  AREA 


89 


FLOOR  PLANS  OF  HOUSE 

a  □ 


1 

LAUNDRY 

COAL    BIN 

1                             I4'6"x    17'6^ 

8'xl4'6" 

8'  X  14' 6" 

1                                                     1    1 

n 

1 

r                          CFI  1  AR 

1    HEATER    J 

L                     hS   22' e" 

16' X  22' 6" 

. 

1 1              1 

a 


□  a 

Basembnt 


D 


D 


90  ELEMENTS  OF  BUSINESS  ARITHMETIC 


PORCH 
6'  6"  X  I  { 


\^ 


V 


KITCHEN 


<■ 


LIBRARY 


13    X    14 


\ 


LIVING    ROOM 


DINING    ROOM 


\ 


<f 


HALL 
9'6"x     11 


PARLOR 


VERANDA  6  6    X    32 


First  Floor 


MEASURES  OF  AREA 


91 


\               PORCH  ROOF               / 

1                       BED     ROOM                   \ 
1                        14' 6"  X    15'                      ^ 

Ir-il 

\ 

L      HALL 

/ 

CLC 

4' 

BED     ROOM 
lo'x    13' 

1 

SET 

x6' 

4'  X  ( 

CLOSE 

1 

6'  6"  X  8' 
BATH     ROOM       \ 

9 

\ 

NTURSLR/ 

lO'  X   lo'e" 

7'  X  8'  6" 



r 

ALCOVE 

A 

BED     ROOM 
13'  X    13' 

1    1 

< 

I 

_DOWN 

k. 

■•                             1     II            ,                  \ 
Q    fi      V      Q                      \ 

P 

/ 

/                                       VERANDA    ROOF                                        \ 
/                                                   7'6"x    34'                                                   \ 

Second  Floor 


92  ELEMENTS  OF  BUSINESS  ARITHMETIC 

4.  A  dining  room  is  18  by  20  ft.  and  11  ft.  high.  At  45^  per  square 
yard  what  will  it  cost  to  lath  and  plaster  the  walls  and  ceiling  ? 

5.  A  bedroom  is  12  ft.  square  and  10  ft.  high.  What  will  it  cost  to 
plaster  it  at  30  f  per  square  yard  ? 

6.  The  rooms  in  the  floor  plan  on  pp.  90-91  are  11  ft.  high.  Making 
allowance  for  doors  and  windows  as  suggested  in  the  discussion,  what 
will  it  cost  to  plaster  the  house,  if  sand  costs  75  f  per  cubic  yard,  lime 
%  1.25  per  barrel,  and  hair  25  ^  per  bushel,  and  the  cost  of  putting  it  on 
is  20  i^  per  square  yard  ? 

7.  What  would  it  cost  to  lath,  and  plaster  the  house  with  stucco 
(900  pounds  to  the  100  square  yards),  if  laths  are  worth  50  ^  a  bundle, 
stucco  $  10.50  per  ton,  and  the  cost  of  putting  on  the  laths  and  stucco  is 
25^  per  square  yard? 

65.  Papering.  Wall  paper  is  put  up  in  single  rolls  8 
yards  long,  and  double  rolls  16  yards  long.  English  rolls 
are  single  rolls  only,  and  are  12  yards  long.  The  width  of 
ordinary  or  figured  paper  is  18  inches,  that  of  ingrain  is  30 
inches.  Higher  grades  vary  somewhat  in  width,  from  18  to 
22  inches,  and  from  30  to  36  inches.  Border  and  friezes  are 
sold  by  the  linear  yard.     They  vary  in  width. 

In  a  roll  of  18-inch  paper,  there  would  be  IJ  rows  of  24 
sq.  ft.,  or  36  sq.  ft.  In  30-inch  paper,  there  would  be  2 J 
rows  of  24  sq.  ft.,  or  60  sq.  ft.  In  22-inch  paper,  there 
would  be  1|  rows,  or  44  sq.  ft.,  etc. 

The  surface  to  be  papered  in  a  room,  18  x  22  ft.,  12  ft. 
high,  would  be  as  follows  :  In  the  four  walls  there  would  be 
12  rows  of  80  (the  perimeter  of  the  room)  sq.  ft.,  or  960 
sq.  ft.,  and  in  the  ceiling  18  rows  of  22  sq.  ft.,  or  396  sq.  ft. 
In  finding  the  number  of  rolls  necessary  to  cover  the  surface, 
the  waste  due  to  matching  must  be  considered,  and  allowance 
made  for  windows  and  doors.  The  latter  is  usually  20  sq.  ft. 
for  each  opening,  counting  archways  and  double  doors  as 
double  openings.  As  whole  rolls  must  be  bought,  it  often 
happens  that  there  is  a  part  of  a  roll  remaining,  sufficient, 
perhaps,  to  cover  waste  due  to  matching,  after  the  exact 


MEASURES  OF  AREA  93 

surface  has  been  provided  for.     If  a  border  were  used,  its 
width  would  be  deducted  from  the  height  of  the  room. 

If  there  were  two  doors  and  three  windows  in  the  above 
room,  100  sq.  ft.  would  be  deducted,  leaving  860  sq.  ft.  in 
the  walls.  There  being  36  sq.  ft.  in  a  roll,  23|  rolls  would 
cover  the  surface;  24  rolls,  at  least,  would  be  required 
Q  roll  only  being  allowed  for  matching).  Whether  this 
would  exactly  cover  the  surface  would  depend  upon  the 
pattern.  For  the  ceiling,  11  rolls  would  exactly  cover  the 
surface,  and  12  would  probably  be  necessary  to  provide  for 
waste  in  matching.  The  border  would  be  80  ft.  or  27  yd. 
long. 

PROBLEMS 

1.  A  room  is  20  by  24  ft.  and  10  ft.  high.  What  will  it  cost  to  paper 
it  with  18-inch  paper  (always  use  single  rolls  unless  otherwise  mentioned) 
at  50  ^  a  roll,  if  there  are  two  doors  and  three  windows  ? 

2.  A  dining  room  is  18  by  24  ft.  and  11  ft.  high.  What  will  it  cost 
to  paper  it  with  paper  22  in.  wide,  worth  90)^  a  double  roll,  if  there  are 
two  doors  and  four  windows  ? 

3.  How  many  rolls  of  paper  30  in.  wide  will  it  take  to  paper  the 
parlor  in  the  plan  on  page  90,  if  a  border  22  in.  wide  is  used  ? 

4.  How  many  rolls,  36  in.  wide,  to  cover  the  dining  room  (p.  90)  if 
a  border  18  in.  wide  is  used  ?  Cost  at  $  1  a  roll  for  body,  60  ;^  a  yard  for 
border,  and  65  j2^  a  roll  for  ceiling  ? 

5.  Find  cost  of  papering  the  library  (p.  90)  with  paper  22  in. 
wide,  at  95^  a  roll,  and  a  border  18  in.  wide  at  95^.  Deduct  for  a 
wainscoting  3  ft.  high  all  around  the  room. 

6.  Find  cost  of  papering  the  kitchen  (p.  90)  with  18-inch  paper  at 
45  ;*  a  roll. 

7.  Find  cost  of  papering  the  halls  of  the  house  (pp.  90-91),  deducting 
for  the  stairway  and  doors,  with  22-inch  paper  costing  $1.10  a  double 
roll.  Use  a  border  of  the  same  width  costing  30  )*  a  yard,  and  18-inch 
paper  for  ceiling  at  40  ^  a  roll. 

8.  Find  cost  of  papering  the  bedrooms  (p.  91)  with  18-inch  paper 
at  35  j^  a  roll,  using  a  border  of  the  same  width  and  costing  20  ^  a  yard 


94  ELEMENTS  OF  BUSINESS  ARITHMETIC 

LAND  MEASURE  AND  SURVEY 

66.  The  Township.  For  purposes  of  public  record,  so 
that  land  ownership  may  be  secure  and  transfers  of  title 
conveniently  and  safely  made,  a  system  of  land  survey  is 
necessary. 

In  the  United  States  prior  to  the  year  1785,  each  tract  of 
land  was  surveyed  and  its  boundary  line  described  as  start- 
ing from  some  natural  object,  thence  a  certain  distance  in 
some  direction,  thence  in  another  direction  to  another  object, 
etc.  The  results  of  such  surveys  were  irregular  shapes  of 
land  tracts  and  often  endless  litigation.  This  method  is  still 
in  use  in  the  Eastern  states. 

By  an  ordinance  of  the  Continental  Congress,  a  system  of 
rectangular  survey  was  adopted  in  1785  for  the  Northwest 
Territory,  and  later  this  was  applied  to  all  the  Central  and 
Western  states. 

By  this  system,  the  land  is  surveyed  into  rectangular 
townships^  six  miles  square,  with  boundary  lines  conforming 
to  the  points  of  the  compass.  To  locate  a  particular  township 
certain  meridians  are  designated  as  principal  meridians  and 
certain  parallels  as  base  lines  or  standard  parallels.  Twenty- 
four  principal  meridians  have  been  established,  the  first  one 
being  in  Ohio  and  the  last  one  in  Oregon.  Parallels  most 
convenient  in  the  survey  of  a  state  are  selected  for  base  lines. 

The  townships  are  numbered,  in  order,  north  and  south 
of  the  base  line,  and  the  particular  row  of  townships  is 
shown  by  the  number  of  rows  or  ranges  east  and  west  of  the 
principal  meridian.  Thus,  T  7  N,  R  4  W,  would  mean  the 
seventh  township  north  of  the  given  hase  line,  in  the  fourth 
range  of  townships  west  of  the  given  principal  meridian. 

Starting  from  the  principal  meridian,  the  surveyors  mark 
off  points  every  six  miles  on  the  base  line.  From  these 
points,  lines  are  run  to  the  north  by  compass.     Because  the 


MEASURES  OF  AREA 


95 


meridians  approach  as  they  near  the  pole,  it  is  clear  that 
these  north  lines  would  soon  be  less  than  six  miles  apart  as 
they  extend  northward.  To  preserve  the  townships  as  nearly 
uniform  in  size  as  possible,  new  parallels  are  run  at  frequent 
intervals  as  correction  lines^  upon  which  the  north  lines  are 
erected  again,  starting  six  miles  apart. 

The  shaded  township   in   the   accompanying   illustration 


Government  Survey 


96 


ELEMENTS  OF  BUSINESS  ARITHMETIC 


would  be  described  as  Township  6  North,  Range  2  East  of 
the  Principal  Meridian. 

67.    Subdivisions  of  the  Township.     The  surveying  unit, 
the  township,  is  six  miles  square,  and  contains  36  sq.  mi. 


6 

5 

4 

3 

2 

•1 

7 

8 

9 

10 

11 

12 

18 

17 

16 

15 

14 

13 

1.9 

20 

21 

22 

23 

24 

30 

29 

28 

27 

26 

25 

31 

32 

33 

34 

35 

36 

A  TOWNSHIF 

These  square  miles  are  called  sections^  and  are  numbered 
from  1  to  36,  beginning  at  the  northeast  corner,  going  west 
on  the  northern  row  of  sections,  east  on  the  second  row,  west 
on  the  third,  etc. 

Since  (Sec.  53)  there  are  640  acres  in  a  square  mile  or 


MEASURES  OF  AREA 


97 


section,  in  half  a  section  there  are  320  acres,  and  160  acres 
in  a  quarter  section.  One  half  of  the  latter  is  called  an 
"eighty."  A  tract  of  land  within  a  section  is  located  by 
the  use  of  fractions  with  descriptive  directions ;  thus,  N W  ^ 
is  the  northwest  quarter  of  the  section,  and  the  S  J  of  NW 
^  is  the  south  half  of  the  northwest  quarter  section. 

LOCATION  OF  SECTIONS 


N.E.  1/4  of 
N.W.  1/i 

N.E.1/; 

8.1/4 

A  Section 


A  complete  description  of  a  tract  of  land  might  be: 
the  NE  ^  of  the  NW  I  sec.  24,  T  12  N,  R  9  E  of  the 
5th  Principal  Meridian,  which  would  be  read  as  the  north- 
east one  fourth  of  the  northwest  quarter  of  section  24, 
township  12  North,  range  9  East  of  the  5th  Principal 
Meridian. 


98  ELEMENTS  OF  BUSINESS  ARITHMETIC 

PROBLEMS 

SUBDIVISION  OF  SECTIONS 

1.  Make  a  diagram  of  a  township.    Locate  Sec.  20,  12,  9,  and  13. 

2.  Make  a  diagram  of  a  section.     Locate  the  NW  ^,  and  S  ^. 

3.  Mr.  Brown  owns  40  acres  in  the  SE  corner  of  section  13,  township 
5  N,  range  3  E.     Locate  his  land. 

4.  A  man  bought  the  E  i  of  the  NW  i,  the  W  J  of  the  NE  \,  and  the 
NE  i  of  the  SE  ^.     How  many  acres  did  he  buy?    Locate. 

5.  I  sold  the  E  ^  of  a  section  of  land  for  $24.50  per  acre,  the  NW  ^ 
for  $  28,  the  E  i  of  SW  I  for  $  23.75,  and  the  remainder  of  the  section 
for  $  20.     How  much  did  I  get  for  the  entire  section  ? 

6.  A  man  bought  the  following  property :  the  SE  I,  the  NE  J,  the 
N  I  of  NW  i,  the  NW  \  of  the  SW  i,  and  the  E  i  of  the  SW  \  of  Sec. 
16,  T  13  N",  R  3  E.     How  many  acres  did  he  buy  ? 

MEASUREMENT  OF  NONRECTANGULAR  SURFACES 

68.  Surface  Forms.  A  surface  is  the  outside  or  face  of  a 
solid. 

A  plane  surface  is  one  that  is  flat.  A  line  connecting  any 
two  points  in  such  a  surface  will  lie  wholly  within  the  sur- 
face. Being  the  outside  or  face  merely,  a  surface  is  not  con- 
sidered as  having  thickness.  It  is  said,  therefore,  to  have 
but  two  dimensions,  length  and  breadth. 

The  edges  of  surfaces  form  lines.  These  lines  may  be 
straight,  when  their  direction  is  never  changed ;  or  curved, 
when  continually  changing,  so  that  no  three  adjacent  points 
are  in  the  same  straight  line. 

An  angle  is  the  difference  in  direction  between  two  straight 
lines  that  meet. 

When  two  meeting  lines  go  in  exactly  opposite  directions, 
the  angle  formed  is  a  straight  angle  (a).     The  angles  formed 

b a c    on  either  side  of  (a),  by  lines  ab 

«  and   ac,  leaving   the   point   a   in 

exactly  opposite  directions,  are  straight  angles. 


MEASURES  OF  AREA 


99 


A  right  angle  is  one  half  a  straight  angle. 
When  one  line  meets  another  so  as  to  form   a 
two   equal    angles,    as    ade    and    adc^   the 
angles  formed  are  right  angles  (5). 

An  acute  angle  is  less  than  a  right  angle,  and  an  ohtu%e 
angle  is  greater  than  a  right  angle  (c?  and  d^. 


c  d 

Lines  are  parallel  when  they  extend  in  the  same  direction 
and  are  at  all  points  equally  distant  from  each  other  (e), 

.        Lines  are  perpendicular  when  the 

angles  formed  by  their  meeting  are 

right   angles    (5)  ;     horizontal  when 
parallel  to  any  line  connecting  two  points  on 
the  horizon  of  vision  (/) ;  vertical  when  per- 
pendicular to  a  horizontal  plane  (5). 

A  triangle  is  a  plane  surface  bounded  by 
three  straight  lines. 

A  triangle  is  equilateral  when  its  three  sides 
are  equal  (/) ;  isosceles  when  two  of  its  sides 
are  equal  (^),  and  scalene  when  all  three  sides  are  unequal  (^). 
A  right   triangle    has 
one  right  angle  (A),  an 
obtuse   triangle  has   one 
obtuse  angle  (^),  and  an 
acute  triangle  has  three 
acute  angles  (/). 

A  polygon  is  a  plane 
9  surface    of    more    than 

four   sides    and   bounded   by   straight   lines. 

The  aggregate  length  of  all  the  lines  bounding  a  polygon 
is  called  its  perimeter. 


100 


ELEMENTS  OF  BUSINESS  ARITHMETIC 


A  quadrilateral  is  a  plane  surface 
bounded  by  four  straight  lines. 

A  parallelogram  is  a  quadrila- 
teral the  opposite  sides  of  which 
are  parallel  (/). 

A  trapezoid  is  a  quadrilateral  two 
sides  of  which  are  parallel  (Ic). 
A  trapezium  is  a  quadrilateral  with  no  two  sides  parallel  (Z). 


h 


A  rectangle  is  a  parallelogram  the  angles   of   w^hich   are 
right  angles  (m), 

A  square  is  a  rectangle,  the  sides  of  which  are  equal  (n). 


ni 


Other  polygons  are  named  likewise  from  the  number  of 
sides,  i.e.  pentagon  (o),  hexa- 
gon   (jo),    octagon,    decagon 
(five,  six,  eight,  ten),  etc. 


MEASURES  OF  AREA 


101 


A  circle  is  a  plane  sur- 
face, every  point  in  the 
boundary  of  which  is 
equidistant  from  a  point 
within  called  the  center. 
The  curved  line  forming 
the  boundary  of  a  circle  is 
called  the  circumference. 
The  distance  from  a  given 
point  on  the  circumfer- 
ence through  the  center  to 
the  opposite  point  is  the 
diameter.  One  half  this 
diameter  is  the  radius  Qq) . 


QUADRILATERALS 

69.  Oblique  Parallelogram.  Take  a  piece  of  paper  and 
make  a  rectangle  3  by  5  in.  Mark  off  one  inch  along  the 
upper  edge  from  the  right  upper  corner.  Draw  a  line  from 
this  mark  to  the  right  lower  corner.  Fold  along  this  line, 
crease,  and  tear  off. 

Along  the  lower  edge  or  base,  mark  off  1  in.  from  the  left 
lower  corner,  and  draw  a  line  to  the  left  upper  corner,  crease, 

and  tear.  The  resultant  figure 
is  an  oblique  parallelogram. 
Why  a  parallelogram?  Why 
oblique  ? 

Again,  on  the  lower  base, 
mark  off  1  in.  from  the  right 
lower  corner,  and  draw  a  line 
to  the  right  upper  corner.  Crease  and  tear.  Place  the 
triangle  to  the  left  edge  of  the  parallelogram  so  that  the 
oblique  edges  meet.  The  oblique  parallelogram  has  been 
changed  into  a  rectangle.     Has   the   base   been   changed? 


102 


ELEMENTS  OF  BUSINESS  ARITHMETIC 


The  altitude  ?     The  perpendicular  height  of  a  figure  is  its 
altitude. 

An  oblique  parallelogram  is  equivalent  to  a  rectangle  with  the 
same  base  and  altitude. 

70.  Trapezoid.  Cut  out  a  5-inch  paper  square.  Two 
inches  to  the  left  of  its  right  upper  corner  draw  a  line  to  the 

right  lower  corner.  Fold  along  this 
line,  crease,  and  tear.  The  four-sided 
piece  is  a  trapezoid.     Why  ? 

Fold  the  trapezoid  so  that  the  parallel 
sides  will  meet.     Crease  and  tear.    Turn 
the  upper  piece  upside  down  and  place 
it  to  the  right  of  the  lower  one,  so  that 
the   oblique    edges  will   meet.       What 
have  you  formed?     Compare  the  base 
of    the    rectangle    with    the    combined 
length  of  the  two  parallel  sides  (called  bases)  of  the  trape- 
zoid.    How  does  its  altitude  compare  with  that  of  the  original 
trapezoid  ? 

A  trapezoid  is  equivalent  to  a  rectangle  with  a  base  equal  to 
the  sum  of  the  two  bases  of  the  trapezoid^  and  one  half  its 
altitude. 

PROBLEMS 

1.  One  side  of  a  walk  is  30  ft.  long  and  the  other  27^  ft.  and  the 
width  5  ft.     How  much  will  it  cost  to  build  it  at  30/  per  square  yard  ? 

2.  One  side  of  a  field  in  the  shape  of  a  trapezoid  measures  180  yd., 
the  other  200  yd.  How  many  acres  in  the  field  if  the  perpendicular  dis- 
tance between  the  sides  is  75  yd.  ? 

3.  What  is  the  area  of  an  oblique  parallelogram,  the  perpendicular 
height  being  24  ft.  and  the  length  56  ft.  ? 

TRIANGLES 

71.  Right  Triangles.  Take  a  piece  of  paper  4  inches  square. 
Fold  and  cut  along  the  diagonal.     You  now  have  two  right 


MEASURES  OF  AREA 


103 


angles.  Take  one  of  them  and  fold  so  that  the  vertex  of  the 
upper  angle  will  meet  the  vertex  of  the  right  angle.  Crease 
and  tear.  Place  the  upper  part 
upside  down  and  with  its  oblique 
side  next  to  the  oblique  side  of  the 
larger  piece.  What  have  you  made  ? 
How  does  the  base  of  the  rectangle 
compare  in  length  with  that  of  the 
triangle  ?  The  height  of  the  rec- 
tangle with  the  altitude  of  the 
triangle  ? 

A  right  triangle  is  equivalent  to  a  rectangle  on  the  same  base 
with  half  the  altitude. 


72.  Isosceles  Triangle.  Take  a  4-inch  paper  square. 
From  the  middle  point  of  the  upper  edge  draw  lines  to  the 

lower  corners.  Fold  along  the  lines, 
crease,  and  tear.  The  large  triangle 
remaining  is  an  isosceles  triangle. 
Why? 

Fold  the  upper  vertex  down  so  it 
will  just  touch  the  middle  point  of 
the  base.  Crease  and  tear.  Con- 
nect the  middle  points  of  the  two 
bases  by  a  straight  line.  Fold  along 
this  line,  crease,  and  tear.  Turn  the  two  pieces  upside  down 
and  place  one  on  each  side  of  the  triangular  piece,  with 
oblique  edges  together.  What  have  you  formed  ?  Compare 
base  of  the  triangle  with  that  of  the  original  triangle.  Com- 
pare their  altitudes. 

An  isosceles  triangle  is  equivalent  to  a  rectangle  on  the  same 
base  and  with  half  the  altitude.  As  an  equilateral  triangle  is 
isosceles,  the  same  would  hold  true. 


104  ELEMENTS  OF  BUSINESS  ARITHMETIC 

73.  Scalene  Triangle.  Cut  a  paper  triangle,  no  two  sides  of 
which  are  equal,  and  not  a  right  triangle.  Fold  over  the 
vertex  to  meet  the  base,  and  so  that  the  line  of  the  fold  will 

be  parallel  with  the  base.  Crease 
and  open.  Fold  again  along  a  line 
drawn  perpendicular  from  the  ver- 
tex to  the  base.  Crease  and  tear. 
Tear  both  new  triangles  at  first  fold. 
Place  the  oblique  sides  of  the  two 
parts  of  the  new  triangles  together. 
Then  place  the  two  resultant  rec- 
tangles side  by  side.  Compare  the  entire  base  with  that  of 
the  original  triangle.  Compare  their  altitudes.  A  scalene 
triangle  is  equivalent  to  a  rectangle  on  the  same  base  and  with 
one  half  the  altitude. 

As  the  same  truth  is  seen,  in  the  above,  to  hold  good  for 
equilateral,  isosceles,  and  scalene  triangles,  whether  right, 
obtuse,  or  acute-angled,  the  general  statement  may  be  made 
that  anT/  triangle  is  equivalent  to  a  rectangle  on  the  same  base 
and  with  half  the  altitude, 

74.  Other  Polygons.  All  other  polygons  may  be  resolved 
into  rectangles  or  triangles.  Finding  the  area  of  the  parts, 
the  whole  may  be  found.  For  example,  among  the  more 
common  geometrical  figures,  a  trapezium  may  be  resolved 
into  two  triangles ;  a  hexagon  and  octagon  may  be  resolved 
into  six  and  eight  triangles  focusing  at  the  center. 

PROBLEMS 

1.  How  many  square  inches  in  a  right  triangle,  if  the  base  is  18  in.  and 
the  altitude  24  in.  ? 

2.  The  base  of  an  isosceles  triangle  is  12  ft.  and  the  altitude  is  16  ft. 
What  is  the  area? 

3.  How  many  square  feet  in  a  scalene  triangle,  with  the  base  15  ft. 
and  altitude  20  ft.  ? 


MEASURES  OF  AREA  105 

4.  A  triangular  field  is  40  rd.  along  its  base  and  26  rd.  along  the  alti- 
tude.    What  is  it  worth  at  $72  per  acre? 

5.  What  is  the  area  of  a  hexagon  composed  of  equilateral  triangles, 
each  side  of  which  is  8  in.  ?     (Use  the  nearest  inch  as  altitude.) 

6.  What  is  the  cost  of  painting  a  church  steeple,  the  base  of  which  is 
an  octagon  8  ft.  on  each  side,  and  whose  slant  height  (altitude)  is  90  ft., 
at  25^  per  square  yard? 

7.  I  have  a  triangular  lawn  that  I  wish  to  sod.  The  base  is  25  yd. 
and  the  altitude  is  24  yd.     What  will  it  cost  at  $1.25  per  square  rod? 

8.  A  hexagonal  silo  is  12  ft.  on  a  side  and  the  distance  from  the 
middle  point  of  a  side  to  the  center  is  10.4  ft.  What  is  the  area  of 
the  bottom  ? 

9.  I  have  a  plot  of  ground  in  the  form  of  a  trapezium.  One  of  the 
diagonals  measures  42  yd.,  and  the  altitudes  of  the  two  triangles  measure 
20  yd.  and  18  yd.  respectively.     What  is  its  area  ? 

75.  Ratio  of  Circumference  of  Circle  to  its  Diameter.  Pro- 
cure a  well-cut  circular  disk,  or  a  wheel.  Measure  carefully 
its  diameter.  Mark  distinctly  some  point  on  the  circumfer- 
ence. Starting  with  that  point  at  the  end  of  a  line  drawn  on 
some  flat  surface,  roll  the  disk  or  the  wheel  along  that  line 
until  the  marked  point  on  the  circumference  again  touches 
the  line.  Measure  the  length  of  the  line  thus  marked  off. 
This  will  be  the  length  of  the  circumference.  Divide  this 
length  by  the  length  of  the  diameter.  The  quotient  or  ratio 
will  be  found  to  be  nearly  3^.  In  other  words,  the  circum- 
ference of  a  circle  is  3^  times  its  diameter.  For  greater 
accuracy,  the  ratio  of  3.1416  is  used.  In  mathematics,  its 
symbol  is  the  Greek  letter  tt  (pronounced  pi).  For  ordinary 
purposes  3^  is  sufficiently  exact. 

76.  Area  of  a  Circle.  Fold  a  rectangular  piece  of  paper, 
5  by  6  inches,  so  that  the  long  edges  meet ;  crease  but  do  not 
open.  Fold  so  that  the  short  edges  meet ;  crease  but  do  not 
open.     Fold  so  that  the  folded  edges  will  meet,  and  crease. 


106  ELEMENTS  OE  BUSINESS  ARITHMETIC 

Mark  off  two  inches  from  the  point  on  both  the  edges,  and 
midway  between  the  edges.  Connect  the  three  points  thus 
made  by  a  uniformly  curved  line.  Cut  through  the  paper 
on  this  line.  Open  the  paper,  and  a  circle  is  shown.  Cut 
through  all  the  creased  lines,  forming  eight  equal  sectors. 
Arrange  seven  of  these  sectors  in  a  row,  with  alternate  points 

and   arcs  along 

C  "7^^  ^TC  7\  ^  one  side.     Fold 

the    oblique 

V       edges   of    the 

\     eighth  sector 

_3  together;  crease 

The  Sectors  of  a  Circle  ^^^  ^^^*      Flace 

a  half  on  each 
end  of  the  row.  What  have  you  approximately  formed 
(see  figure)?  What  part  of  the  circumference  forms  the 
base  ?  What  is  the  altitude  ?  If  the  sectors  were  sixteenths, 
the  resultant  would  be  more  nearly  a  perfect  rectangle  ;  if 
thirty-seconds,  still  more  nearly,  etc.  We  may  conclude, 
then,  that : 

A  circle  is  equivalent  to  a  rectangle  with  one  half  the  circum- 
ference as  a  base,  and  the  radius  as  its  altitude. 

Note.  —  The  geometrical  formula  for  the  area  of  a  circle  is  ttt^,  or  the 
square  of  the  radius  multiplied  by  3^. 

PROBLEMS 

1.  The  area  of  a  circle  14  in.  in  diameter  with  a  circumference  of 
44  in.  is  equivalent  to  a  rectangle  with  a  base  of inches  Q  the  cir- 
cumference), and  an  altitude  of inches  (radius). 

2.  What  are  the  dimensions  of  a  rectangle  that  may  be  formed  from  a 
circle  7  in.  in  diameter  and  a  circumference  of  22  in.  ? 

3.  The  diameter  of  a  circle  is  21  in.,  the  circumference  66  in.  Find 
the  area. 

4.  Find  the  area  of  a  circle  with  a  diameter  of  8  in. 


MEASURES  OF  AREA  107 

OTHER  SURFACES 

If  a  paper  6  by  8  inches  were  rolled  into  a  cylinder  with 
the  long  edges  meeting,  what  would  be  the  circumference  of 
the  cylinder  ?     What  the  length  ? 

The  surface  of  a  cylinder  is  equivalent  in  area  to  that  of  a 
rectangle  having  the  length  of  the  cylinder  as  a  hase^  and  its  cir- 
cumference as  the  altitude. 

The  surface  of  a  pyramid  consists  of  the  rectangular  base 
and  triangular  sides.  The  triangles  have  a  common  altitude,  or 
slant  height,  hence  the  lateral  area  of  a  pyramid  is  equivalent  to 
that  of  a  rectangle  having  the  perimeter  of  the  pyramid  as  a  base, 
and  one  half  the  slant  height  as  the  altitude. 

The  lateral  area.,  likewise,  of  a  cone  is  equivalent  to  that  of  a 
rectangle  having  the  circumference  of  the  cone  as  a  hase^  and 
one  half  the  slant  height  as  the  altitude. 

The  surface  of  a  sphere  is  equivalent  to  a  rectangle  having  the 
circumference  of  the  sphere  for  a  hase^  and  its  diameter  for  the 
altitude. 

GENERAL    PROBLEMS 

1.  What  is  the  area  of  the  surface  of  a  globe  6  in.  in  diameter  ? 

2.  What  is  the  entire  surface  of  a  cylinder  whose  diameter  is  6  in., 
height  2  f  t.  ? 

3.  A  cylindrical  cistern  is  10  ft.  deep  and  7  ft.  in  diameter.  What 
will  it  cost  to  cement  it,  at  35  /'  a  square  yard  ? 

4.  In  a  warehouse  there  are  six  hoppers  of  pyramidal  shape,  6  ft. 
high  and  8  ft.  square.     How  many  square  feet  on  their  surfaces? 

5.  What  will  it  cost,  at  24-  ^  per  square  yard,  to  paint  a  spire  of  conical 
shape,  if  the  slant  height  is  75  ft.  and  the  diameter  of  the  base  is  16  ft.  ? 

6.  A  rectangular  park  is  80  by  160  ft.  In  the  left-  and  right-hand 
corners  at  one  end  are  flower  beds  which  are  5  ft.  wide,  and  extend  15  ft. 
both  ways  from  the  corners.  In  the  left-hand  corner  at  the  other  end  is 
a  triangular  bed,  the  altitude  of  which  is  12  ft.  and  the  base  16  ft.  In 
the  right-hand  corner  is  a  bed  in  the  form  of  a  trapezoid  7  ft.  broad, 
whose  sides  are  respectively  24  and  18  ft.  In  the  center  is  a  pond  42  ft. 
in  diameter,  with  a  walk  3  ft.  wide  extending  from  one  side  of  the  park 


108  ELEMENTS  OF  BUSINESS  ARITHMETIC 

to  and  around  the  pond.     The  remaining  space  is  a  lawn.     How  many 
square  feet  in  the  lawn?    In  each  bed?    In  the  pond?    In  the  walk? 

7.  A  street  in  a  city  is  100  ft.  wide  and  1  mi.  long.  What  will  it 
cost  to  have  it  paved  with  brick,  4  by  8  in.,  at  $  6.20  per  M.  ? 

8.  A  horseman  has  a  field  20  rd.  square  in  which  he  puts  a  circular 
track  50  ft.  wide.     What  is  the  area  of  the  track  ?     Its  length  ? 

9.  A  man  owned  a  lot  80  rd.  square.  A  corporation  bought  the  right 
of  way  for  an  electric  line  at  $195  per  acre.  One  fence  inclosing  the 
track  was  to  commence  at  the  northwest  corner  of  the  lot  and  run  to  a 
point  20  rd.  east  of  the  southwest  corner ;  the  second  fence  to  commence 
at  a  point  15  rd.  south  of  the  northeast  corner  and  run  to  a  point  12  rd. 
east  of  the  first.  What  did  the  right  of  way  cost?  How  many  posts, 
set  8  ft.  apart,  will  it  take  for  the  fences?  Find  cost  of  the  fencing  at 
65  ^  a  rod. 

10.  A  room  is  15  by  18  ft.  How  many  yards  of  carpet,  27  in.  wide, 
will  it  take  to  cover  it,  allowing  9  in.  for  matching? 

11.  A  building  is  40  by  80  ft.  and  15  ft.  high.  The  altitude  of  the 
gable  ends  is  14  ft.,  and  the  roof  projects  2  ft.  over  each  side  and  end. 
What  will  it  cost  to  cover  this  roof  with  shingles,  laid  4^  in.  to  the 
weather,  at  |4.50  per  M?  What  will  it  cost  to  paint  the  roof  and 
the  body  of  the  building,  at  25  f  per  square  yard  ? 

12.  A  buys  a  piece  of  land  20  rd.  square,  paying  $250  an  acre.  He 
divides  it  into  lots  46  by  100  ft.,  after  deducting  for  a  street  50  ft.  wide 
around  the  piece  and  an  alley  30  ft.  wide  through  the  center.  The  ex- 
pense of  setting  out  trees,  grading,  etc.,  was  $  754.75.  If  he  sold  the 
corner  lots  for  %  250  apiece,  and  the  others  for  %  200  apiece,  what  did  he 
gain  by  the  transaction  ? 

13.  A  house  has  8  rooms,  with  dimensions  as  follows :  3  rooms  15  by 
18  ft.,  2  rooms  10  by  14  ft.,  and  3  rooms  12  by  16  ft.,  and  8,  9,  and  10  ft. 
high  respectively.  There  are  24  windows  and  8  doors.  Find  the  cost  of 
lath  to  cover  them  at  30  ^  a  bundle.  How  much  lime,  sand,  and  hair  will 
it  take  to  plaster  the  house  ?  How  much  stucco  would  it  take,  counting 
900  lb.  to  100  square  yards  ? 

14.  The  first  three  rooms  in  Problem  13  are  to  be  covered  with  18-inch 
paper,  the  next  three  with  22-inch  paper,  and  the  last  two  with  30-inch 
paper.  Allowing  20  sq.  ft.  for  openings,  how  many  rolls  of  each  kind  of 
paper  will  it  take  to  cover  the  rooms,  if  the  doors  and  windows  are 
divided  equally  among  the  rooms  ? 


VIII 

MEASURES  OF  VOLUME 

77.  Units  of  Volume.  Volume  is  the  quantity  of  space  oc- 
cupied. Any  given  space  has  length,  breadth,  and  thick- 
ness. Therefore  volume  has  to  do  with  these  three 
dimensions.  Volume  is  measured  by  cubic  units,  i.e.  units 
whose  length,  breadth,  and  thickness  are  the  same. 

A  rectangular  body  whose  dimensions  are  equal  is  called 
a  cube.  These  cubic  units  correspond  to  units  of  length,  viz. 
a  cubic  inch,  a  cubic  foot,  and  a  cubic  yard. 

The  standard  unit  is  the  cubic  yard.  From  the  fact  that 
volume  is  measured  by  cubic  units,  measures  of  volume  are 
called  cubic  measure. 

Table  of  Cubic  Units 
There  are : 

1728  cubic  inches  (cu.  in.)  in  1  cubic  foot  (cu.  ft.). 
27  cubic  feet  in  1  cubic  yard  (cu.  yd.). 

EXERCISES 

1.  Procure  or  make  a  number  "of  inch  cubes.  Through  touch  and 
sight,  seek  to  develop  an  accurate  concept  of  a  cubic  inch.  Cut,  without 
measurement,  an  inch  cube  from  any  available  material.  Test  its  accuracy. 
Practice. 

2.  Practice  estimating  the  volume  in  cubic  inches  of  a  box,  a  book, 
the  desk  top,  a  piece  of  wood,  etc.  Use  a  ruler  for  testing  the  accuracy 
of  your  judgment. 

3.  By  pasting  the  edges  of  six  one-foot  squares  of  pasteboard  together, 
a  cubic  foot  may  be  made. 

109 


110 


ELEMENTS  OF  BUSINESS  ARITHMETIC 


s.    \. 


4.  Develop  a  concept  of  the  one-foot  cube  in  the  same  manner  as 
above. 

5.  Estimate  the  volume  in  cubic  feet  of  large  boxes,  rooms,  halls; 
foundations  of  buildings,  etc.  Test  correctness  with  a  ruler.  Practice  for 
accuracy  in  judgment  of  the  number  of  cubic  feet  in  different  volumes. 

6.  Measure  oif  a  cubic  yard  from  the  corner  of  a  room.  Fix  as  accu- 
rate a  concept  of  it  as  possible. 

7.  Estimate  volume  of  various  solids  in  cubic  yards.    Test  in  practice. 

78.  Finding  Volume.  Finding  the  volume  of  a  solid  con- 
sists in  finding  the  cubic  units  of  space  occupied  by  it.  In 
finding  the  volume  of  rectangular  solids,  we  first  think  or 

image  tlie  cubic 
unit,  then  the 
number  of  these 
units  along  one 
edge  or  in  one 
row,  then  the 
number  of  such 
rows  forming  a 
layer,  and  finally 
the  number  of 
layers  of  cubic 
units  in  the  solid.  The  number  of  cubic  units  in  one  row  is 
the  same  as  the  number  of  linear  units  along  an  edge  or  one 
dimension;  the  number  of  such  rows  is  the  same  as  the 
number  of  linear  units  along  an  edge  forming  another 
dimension ;  and  the  number  of  layers  is  equal  to  the  linear 
units  along  the  edge  forming  the  third  dimension.  We  say, 
then,  that  the  length  indicates  the  number  of  units  in  one 
row,  the  breadth  the  number  of  rows,  and  the  thickness  the 
number  of  layers. 

To  find  the  volume  of  any  rectangular  solid,  multiply  the 
number  of  cubic  units  in  one  row  by  the  number  of  rows  form- 
ing one  layer^  and  that  product  by  the  number  of  such  layers. 


(a)  A  cubic  unit.     (6)  A  row  of  four  cubic  units,     (c)  A 
layer  of  8  cubic  units,     (d)  A  solid  of  16  cubic  units. 


MEASURES  OF  VOLUME  111 

Thus,  a  solid  6  in.  long,  5  in.  wide,  and  4  in.  high  has  6  cu. 
in.  in  each  row,  30  cu.  in.  in  five  rows,  or  one  layer,  and  120  cu. 
in.  in  the  four  layers  of  the  solid.  This  may  be  expressed  as 
follows : 

6  cu.  in.  X  5  X  4  =  120  cu.  in. 

PROBLEMS 

1.  A  box  is  10  by  20  inches  and  4  inches  deep.  How  many  cubic 
inches  in  each  layer  ?     How  many  in  the  box  ? 

2.  A  box  is  6  feet  long,  4  feet  wide,  and  4  feet  deep.  How  many 
layers  of  cubic  feet?    How  many  cubic  feet  in  the  box? 

3.  How  many  cubic  feet  of  water  will  a  tank  hold  if  it  is  12  by  6  feet 
and  3  feet  deep  ? 

4.  What  will  it  cost  to  dig  a  cellar  22  by  16  feet  and  10  feet  deep,  at 
$1.35  a  load  (cubic  yard)? 

5.  How  many  cubic  feet  in  a  piece  of  timber  6"  x  8"  x  40'  ? 

6.  How  many  cubic  feet  of  wheat  in  a  bin  6  by  12  by  8  feet  high  ? 

7.  A  schoolroom  is  60  feet  long,  40  feet  wide,  and  12  feet  high.  If 
there  are  120  pupils,  how  many  cubic  feet  of  air  to  each  pupil?  How 
often  must  the  air  be  changed  to  give  fresh  air,  if  a  pupil  requires  2400 
cubic  feet  of  air  per  hour? 

8.  How  many  cubic  yards  of  earth  must  be  removed  in  digging  a 
trench  for  the  walls  of  a  building,  if  the  building  is  40  by  100  feet,  the 
walls  2  feet  wide,  and  the  trench  3  feet  deep? 

9.  I  wish  to  build  a  road.  The  grade  would  cost  $  80  a  mile.  The 
gravel  is  to  be  spread  9  inches  deep  and  16  feet  wide.  What  will  the 
road  cost  per  mile,  if  the  gravel  is  worth  $  2  per  cubic  yard? 

10.  My  lawn  is  70  by  120  feet.  What  will  be  the  cost  for  dirt  to  raise 
it  18  inches  at  50^  a  load? 

11.  The  abutments  of  a  bridge  are  30  feet  long  and  6  feet  wide  and 
15  feet  high.  Find  the  pressure  exerted,  counting  160  pounds  to  the 
cubic  foot. 

12.  A  man  has  a  pond  5  by  20  rods.  The  ice  is  9  inches  deep.  Count- 
ing 1  of  the  space  for  sawdust,  and  ^^r  loss  of  ice  in  handling,  what  will  be 
the  height  of  a  building  80  feet  long  and  40  feet  wide  that  will  hold  it  ? 


112  ELEMENTS  OF  BUSINESS  ARITHMETIC 

79.  Cubes  and  Roots.  A  cube  as  defined  in  Sec.  77  is  a 
rectangular  body  whose  three  dimensions  are  equal.  That 
is  to  say,  the '  number  of  cubic  units  along  one  edge  is  the 
same  as  the  number  of  rows  and  the  number  of  layers  of 
such  units.  In  finding  the  volume  of  such  a  body,  the  same 
number  will,  therefore,  be  used  as  a  factor  three  times,  and 
the  resultant  volume  is  said  to  be  expressed  by  the  cube  of 
the  number,  or  the  number  is  said  to  be  raised  to  the  third 
power.  Thus,  the  cube  of  three  may  be  expressed :  3  cubic 
units  X  3  X  3  =  27  cubic  units,  or,  without  reference  to  defi- 
nite units  ;  as,  3  x  3  x  3  =  27.  This  may  also  be  expressed 
as  33=  27. 

Finding  the  volume  of  a  cube  when  the  number  of  cubic 
units  along  one  edge  only  is  given,  is  called  cubing  a  number. 
When  the  volume  is  known,  and  the  problem  is  to  find  the 
number  of  cubic  units  along  one  edge,  the  process  is  known 
as  finding  the  euhe  root.  Thus,  the  cube  of  2-  is  8,  or  of  2 
cu.  ft.  is  8  cu.  ft.  The  cube  root  of  8  is  2,  or  the  cube  root 
of  8  cu.  ft.  is  2  cu.  ft. 

EXERCISES 

1.  The  cube  of  three  feet  has  how  many  cubes  in  each  row?  How 
many  rows?    How  many  layers?    How  many  cubic  feet  in  all? 

2.  The  cube  of  5  feet  contains  how  many  rows  ?  How  many  cubic 
feet? 

3.  A  cube  of  9  inches  has  how  many  cubic  inches  ? 

4.  A  cube  containing  125  cu.  ft.  has  how  many  cubic  feet  along  one 
edge? 

5.  A  cube  containing  27  cubic  inches  has  how  many  cubic  inches  in 
each  row  ?    The  cube  root  of  27  is  ? 

6.  The  cube  of  4  is?    Of  6  is?    Of  7  is?     Of  8  is? 

7.  The  cube  of  10  is  ?    Of  11  is  ?    Of  12  is  ?    Of  15  is  ? 

8.  The  cube  of  20  is?  Of  25  is?  Of  30  is?  Of  50  is?  Of  40  is? 
Of  75  is? 

9.  The  cube  root  of  64  is?    Of  27  is?    Of  125  is? 


MEASURES  OF  VOLUME  113 

10.  The  cube  r9ot  of  1000?    Of  1728?    Of  1331?    Of  3375? 

11.  The  cube  root  of  8000?    Of  15,625?    Of  27,000?    Of  1,000,000? 

Memorize  the  following  table  : 


13=1 

73  =  343 

153=3375 

23=8 

83  =  512 

203  =  8000 

33=27 

93  =  729 

253=15,625 

43=64 

103  =  1000 

503  =  125,000 

53  =  125 

113  3=  1331 

1003  =  1,000,000 

63=216 

123  =  1728 

APPLICATIONS  OP  VOLUME 

80.  Wood  Measure.  Volume  measurement  forms  a  con- 
venient way  of  finding  the  amount  of  many  things  bought 
and  sold  in  bulk.  Among  the  simplest  of  its  applications 
is  wood  measure.  For  this  purpose  a  special  unit  is  used. 
Wood  used  for  fuel,  when  first  prepared  for  sale,  is  usually 
cut  4  feet  long.  A  pile  of  wood,  4  feet  high  and  8  feet  long, 
is  called  a  cord.  There  would  be,  then,  4  layers  of  32  cubic 
feet,  or  128  cubic  feet  in  one  cord.  Rough  timber  cut  in 
4-foot  lengths  is  called  cordwood.  A  cord  foot  is  4  feet 
wide,  4  feet  high,  and  1  foot  long,  and  contains  16  cubic 
feet.     There  are  8  cord  feet  in  a  cord. 

81.  Stone  Measure.  Stone  is  divided  into  two  classes  at 
the  quarry,  viz.  dimension  stone  and  rubble.  The  first  con- 
sists of  pieces  which  are  quarried  in  regular  shapes  and  to 
a  fixed  size.  This  class  of  stone  is  generally  sold  by  the 
cubic  foot. 

Rubble  is  the  waste  from  quarrying  the  larger  stones,  and 
includes  pieces  of  various  sizes  and  shapes.  It  is  usually 
sold  by  the  carload,  or  in  small  quantities  by  the  perch  or 
cord,  and  in  some  places  by  the  ton. 

In  the  wall,  dimension  stone  is  generally  measured  by  the 
square  foot.     Rubble  is  measured  by  the  perch,  which  varies 


114  ELEMENTS  OF  BUSINESS  ARITHMETIC 

from  16  to  25  cu.  ft.,  usually  24f  cu.  ft.;  or  the  cord,  of  128 
cu.  ft.,  or  the  cubic  yard.  About  I  of  the  volume,  is  usually 
allowed  for  mortar,  which  is  commonly  mixed  in  the  propor- 
tion of  3  parts  of  sand  to  1  of  lime. 

Walls  of  the  same  thickness  and  height  are  thought  of 
as  one  long  wall,  using  the  outside  measurements  of  the 
building  to  determine  its  length.  No  allowance  is  made  for 
openings  in  estimating  cost  of  labor,  unless  openings  are 
many  or  very  large,  when  |  the  volume  is  sometimes 
deducted. 

82.  Bricklaying.  In  estimating  the  number  of  bricks  for 
a  given  foundation  or  building,  the  walls  are  considered  as 
one  wall,  with  the  outside  perimeter  of  the  four  walls  as  the 
length.  The  height  being  known,  the  area  of  the  side  of 
such  a  wall  is  then  found.  If  the  wall  is  to  be  a  single  brick 
in  thickness,  7J  bricks  will  be  required  for  each  square  foot ; 
if  2  bricks  thick,  15 ;  and  if  3  bricks  thick,  22|-  bricks.  In 
estimating  the  cost  of  labor,  allowance  for  openings  is  not 
usually  made  unless  these  are  many  and  large.  When 
deduction  is  made  for  openings,  it  is  for  one  half  their 
volume. 

The  above  estimates  are  based  on  the  size  of  the  common 
brick,  which  is  about  2x4x8  inches.  As  the  bricks  are 
laid  flat,  the  exposed  side  of  each  brick  would  be  2  x  8  inches, 
or  16  square  inches.  Nine  bricks,  if  laid  without  mortar, 
would  be  required  for  one  square  foot  (16  sq.  in.  x  9  =  144 
sq.  in.).  Allowing  about  -^  for  mortar,  7 J  bricks  would  be 
required. 

Bricks  are  now  made  in  such  a  variety  of  sizes  that  it  is 
often  necessary  to  figure  the  number  to  a  square  foot,  from 
the  known  size  of  the  brick  to  be  used  in  the  particular 
building. 

From  1  to  IJ  barrels  of  lime,  according  to  quality,  and  |- 


MEASURES  OF  VOLUME  115 

of  a  cubic  yard  of  sand  are  required  for  each  1000  bricks. 
No  allowance,  as  a  rule,  is  made  for  openings  in  estimating 
labor.  Four  bricklayers  and  one  tender  will  lay  from  7200 
to  8000  bricks  per  day. 

PROBLEMS 

1.  How  many  cords  of  wood  in  a  pile  24  ft.  long,  8  ft.  high,  and  4  ft. 
wide? 

2.  How  many  cords  of  stone  in  a  wall  8  by  60  ft.  and  2  ft.  thick? 

3.  What  will  a  pile  of  wood  4  by  6  by  48  ft.  cost,  at  $4  per  cord? 

4.  What  will  it  cost  to  build  the  stone  foundations  for  a  barn,  20  by 
40  ft.,  if  the  wall  is  6  ft.  high  and  2  ft.  thick,  at  $  15  per  cord? 

5.  If  a  cellar  is  18  by  36  ft.  and  the  walls  10  ft.  high,  how  many 
bricks  will  it -take  to  build  it  two  bricks  thick? 

6.  How  many  bricks  would  be  necessary  for  the  foundations  of  a 
building  25  by  30  ft.,  if  it  is  4  ft.  high  and  3  bricks  thick? 

7.  The  foundation  of  a  house  is  3  bricks  thick  and  3  ft.  high.  The 
total  length,  if  extended,  is  175  ft.  How  many  bricks  will  it  take?  How 
much  lime  and  sand  ?  What  will  be  the  cost  of  the  bricks  at  $  9.50  per 
M  ?  How  long  will  it  take  four  bricklayers  and  one  tender  to  lay  the 
foundation  ? 

8.  Commencing  at  the  northwest  corner,  a  wall  extends  east  30  ft., 
then  south  10  ft.,  then  west  10  ft.,  then  south  10  ft.,  then  east  10  ft., 
then  south  10  ft.,  then  west  15  ft.,  then  north  5  ft.,  then  west  15  ft.,  then 
north  to  the  point  of  starting.  The  wall  is  3  bricks  thick  and  4  ft.  high. 
How  many  bricks  will  it  take  to  lay  it?     How  much  sand  and  lime? 

9.  In  the  floor  plan  on  p.  88,  how  many  cords  of  stone  would  it 
take  to  lay  the  foundation  18  in.  thick  and  7  ft.  high?  How  many  cubic 
yards?  How  many  perch  of  24|  cu.  ft.?  How  much  lime  and  sand 
would  it  take?  How  many  bricks  would  it  take  for  a  wall  3  bricks 
thick?  What  will  be  the  cost  of  brick  at  $  8.75  per  M?  What  will  be 
the  cost  of  lime  and  sand  for  either  brick  or  stone  wall,  the  lime  at 
$  1.00  per  barrel ;  sand  at  75^  a  load  of  one  cubic  yard? 

83.  Board  Measure.  Lumber  is  measured  by  the  hoard 
foot.     One  square  foot  of  surface  on  a  board  one  inch  thick 


116  ELEMENTS  OF  BUSINESS  ARITHMETIC 

or  less  is  called  a  board  foot.  A  board  foot  is  added  for 
each  added  inch  in  thickness. 

According  to  a  method  used  by  large  dealers,  the  board 
12'  long  and  12' '  wide  is  used  as  a  basis  for  computation. 
The  total  number  of  board  feet  in  the  given  number  of 
boards  is  increased  or  decreased  by  such  a  fraction  of  the 
result  as  the  length  is  of  12'  and  the  width  is  of  12".  A 
board  12'  long  and  12"  wide  would  contain,  manifestly,  12 
board  feet.  A  board  6"  wide  would  contain  ^  as  much,  or 
6  board  feet;  8"  wide,  f  of  12  board  feet,  or  8  board  feet,  etc. 

Twelve-foot  boards,  then,  contain  as  many  hoard  feet  as  the 
hoard  is  inches  wide.  Thus,  24  such  boards  8"  wide  would 
contain  24  times  8  board  feet,  or  1092  board  feet ;  if  14-foot 
boards,  they  would  contain  \  more,  or  1274  board  feet ;  if 
16'  long,  \  more,  or  1456  board  feet.  The  width  of  boards 
is  counted  as  inches  and  half  inches.  If  a  board  is  not  an 
even  size,  it  is  considered  as  the  next  smaller  half  inch. 

84.  Log  Measure.  For  estimating  the  amount  of  lumber 
that  may  be  cut  from  a  given  log,  or  for  estimating  the  value 
of  logs  to  be  sold,  it  is  necessary  first  to  find  a  mean  diameter. 
This  is  found  by  taking  \  of  the  sum  of  the  diameters  of  the 
two  ends  of  the  log,  inside  the  bark. 

Slabs  are  taken  off  to  get  the  log  into  rectangular  form 
and  wide  enough  to  be  of  use.  4"  of  the  diameter  (2"  on 
each  of  the  four  sides)  is  deducted  for  slabs,  and  \  of  the 
solid  contents  for  waste  in  sawing  (kerf)  and  in  edging. 
Thus,  a  log  with  an  average  diameter  of  14",  12'  long,  would 
form  a  rectangular  log  10"  by  10",  12'  long,  and  would  con- 
tain 100  board  feet.  Allowing  \  for  kerf  and  other  waste, 
the  board  product  would  be  75  board  feet.  If  the  length  of 
the  log  was  16',  there  would  be  \  more,  or  100  board  feet. 
Computations  for  other  lengths  are  made  in  the  same  way. 
(Sec.  83.) 


MEASURES  OF  VOLUME 


117 


The  accompanying  diagrams  show  two  of  the  many  ways 
of  sawing  logs. 


85.  Lumberman's  Reference  Table.  The  following  table, 
similar  in  form  to  the  Lumberman's  Reference  Table,  may 
be  used  for  drill  in  finding  mentally  the  number  of  board 
feet  in  boards,  joists,  etc.,  of  various  dimensions,  using  the 
table  for  verifying  results.  Notice  that  if  the  answer  con- 
tains a  fraction  less  than  |-,  it  is  dropped,  while  if  it  is  more 
than  J,  another  foot  is  added. 

PROBLEMS 

1.  How  many  board  feet  in  50  boards  8  inches  wide  and  12  feet  long? 

2.  How  many  board  feet  in  60  boards  10  inches  wide  and  12  feet  long? 

3.  How  many  board  feet  in  10  boards  2  x  4-12  ? 

4.  How  many  board  feet  in  20  boards  3  x  6-16  ? 

5.  How  many  board  feet  in  60  boards  1  x  10-20  ? 

6.  How  many  board  feet  in  10  boards  6  x  6-24  ? 

7.  The  Koontz  Lumber  Company  sold  the  following  bill  of  lumber : 
20  pc.  2x6-10,  15  pc.  4x4-14,  25  pc.  4x4-16,  40  pc.  2x6-14,  75  pc. 
2  x  10- 8  at  $20  per  M ;  100  po.  2  x  10-20,  50  pb.  2  x  6-22,  45  pc.  2  x  8-18, 
10  pc.  2  X  12-16,  75  pc.  2  x  8-18,  at  $22  per  M ;  2050  posts  at  $24  per  C  ; 
25  pc.  6x8-24  at  $20  per  M;  125  bunches  shingles  at  $4.50  per  M. 
Find  the  amount  of  the  bill. 


118 


ELEMENTS  OF  BUSINESS  ARITHMETIC 


Eeference  Table 


BOARD  MEASURE,  JOISTS,  SCANTLING,  AND  TIMBER 

Size 

12 

14 

16 

18 

20 

22 

24 

26 

28 

30 

2x4 

8 

9 

11 

12 

13 

15 

16 

17 

19 

20 

2x6 

12 

14 

16 

18 

20 

22 

24 

26 

28 

30 

2x8 

16 

19 

21 

24 

27 

29 

32 

35 

37 

40 

2x10 

20 

.  23 

27 

30 

33 

37 

40 

43 

47 

50 

2x12 

24 

28 

32 

36 

40 

44 

48 

52 

56 

60 

3x6 

18 

21 

24 

27 

30 

33 

36 

39 

42 

45 

3x8 

24 

28 

32 

36 

40 

44 

48 

52 

56 

60 

3x10 

30 

35 

40 

45 

50 

55 

60 

65 

70 

75 

3x12 

36 

42 

48 

54 

60 

66 

72 

78 

84 

90 

3x14 

42 

49 

56 

63 

70 

77 

84 

91 

98 

105 

4x4 

16 

19 

21 

24 

27 

29 

32 

35 

37 

40 

4x6 

24 

28 

32 

36 

40 

44 

48 

52 

56 

60 

4x8 

32 

37 

43 

48 

53 

59 

64 

69 

75 

80 

6x6 

36 

42 

48 

54 

60 

66 

72 

78 

84 

90 

6x8 

48 

56 

64 

72 

80 

88 

96 

104 

112 

120 

6x10 

60 

70 

80 

90 

100 

110 

120 

130 

140 

150 

8x8 

64 

75 

85 

96 

107 

117 

128 

139 

149 

160 

8x10 

80 

93 

107 

120 

133 

147 

160 

173 

187 

200 

8x12 

96 

112 

128 

144 

160 

176 

192 

208 

224 

240 

10x10 

100 

117 

133 

150 

167 

183 

200 

217 

233 

250 

10x12 

120 

140 

160 

180 

200 

220 

240 

260 

280 

300 

12x12 

144 

168 

192 

216 

240 

264 

288 

312 

336 

360 

12x14 

168 

196 

224 

252 

280 

308 

336 

364 

392 

420 

14x14 

196 

229 

261 

294 

327 

359 

392 

425 

457 

490 

8.  How  many  board  feet  in  a  log  2  ft.  in  diameter  at  one  end  and 
2'  6"  at  the  other,  and  12  ft.  long? 

9.  A  tree  is  60  in.  in  circumference  at  the  base  and  48  in.  at  the  top, 
and  42  ft.  high.     How  many  feet  of  lumber  may  be  sawed  from  it? 

10.   A  man  built  a  close  board  fence  4  ft.  high  around  a  lot  120  yd. 
square.     The  boards  were  nailed  in  upright  position  to  two  railings  2  by 


MEASURES  OF  VOLUME  119 

4  in.,  and  the  posts  were  set  8  ft.  apart.  He  paid  |20  per  M  for  the 
railings,  $30  per  C  for  the  posts,  and  |25  per  M  for  the  boards.  What 
did  the  material  cost  ? 

11.  How  many  feet  of  lumber  will  it  take  to  board  a  barn  40  by  80 
ft.  and  25  ft.  high,  if  the  distance  from  the  girders  to  the  ridge  of  the 
roof  is  18  ft.  ?    Cost  at  $  32  per  M  ? 

12.  A  lot  50  by  160  ft.  is  inclosed  by  a  picket  fence.  The  pickets  are 
4  ft.  long,  3  in.  wide,  and  1  in.  thick,  and  are  placed  3  in.  apart.  The 
posts  are  placed  8  ft.  apart,  2  x  d's  being  used  at  top  and  bottom  for  railing, 
and  a  baseboard  10  in.  wide.  What  will  the  fence  cost,  if  the  posls  are 
$20  per  C,  the  lumber  $22  per  M,  and  pickets  $4.25  per  C  ? 

13.  Mr.  A.  H.  Ward  buys  the  following  bill  for  a  small  house  :  1  pc. 
6x8-20,  2  pc.  2x8-26,  1  pc.  2 x  8-20,  4  pc.  2x8- 18  @$26;  45  pc. 
2  X  8  - 14,  34  pc.  2  X  8  - 12,  44  pc.  2  x  6  - 12,  38  pc.  2x6-  12,  3  pc.  2  x  6  - 
18@$23;  130pc.  2x4-18,  30  pc.  2x4-16,  30  pc.  2x4-14,  50  pc.2x 
4  -  12  @  $28 ;  1700  ft.  #  1  sheathing,  10,000  ft.  #  1  fence  flooring,  2000  ft. 
#  shiplap,  2000  ft.  #  2  fence  flooring,  700  ft.  straight  grain  flooring,  1360 
ft.  cedar  siding  @  $24;  320  ft.  polished  brass  ceiling,  20  pc.  1x6-1.5, 
G.  P.  finishing  @  $  28;  5  rolls  best  paper,  2500  ft.  @  $  1.50  per  C  ;  7  win- 
dows, 10x16  @  $1.80;  3  doors,  2  ft.,  8  in.  by  6  ft.  8  in.  @  $5;  10  bbl. 
lime  @  $1.15;  5000  hard  brick  @  $9.25  per  M;  200  ft.  3^  in.  crown 
molding  @  $  2..50  per  hundred  linear  feet ;  200  ft.  2-in.  crown  molding  @ 
$  2.50  per  hundred  linear  feet.     What  is  the  amount  of  the  bill  ? 

MEASURES  OF  BULK 

86.  Dry  Measure.  For  measuring  small  solids,  such  as 
grains,  fruits,  vegetables,  etc.,  vessels  are  used  whose  volume 
or  capacity  is  known.  The  volume  of  liquids  must  be  ob- 
tained in  the  same  way.  Inasmuch  as  the  solid  form  and  the 
varying  shape  of  grains,  fruits,  etc.,  leave  unoccupied  spaces 
between  the  solids,  while  liquids  are  compact,  different  unit 
measures  have  been  established.  The  units  for  measuring 
solids  form  what  is  known  as  dry  measure. 

The  established  unit  for  dry  measure  is  the  bushel.  It  is 
usually  cylindrical  in  form,  18^  inches  in  diameter,  and  8 
inches  deep,  containing  2150.42  cubic  inches.  For  grain, 
shelled  corn,  etc.,  these  measures  are  filled  to  the  level,  called 


120  ELEMENTS  OF  BUSINESS  ARITHMETIC 

"stricken  measure."  For  apples,  potatoes,  ear  corn,  etc., 
the  measure  is  "heaped,"  and  the  bushel  is  supposed  to 
contain  2747.71  cubic  inches. 

For  finding  the  capacity,  in  bushels,  of  a  grain  bin,  granary, 
or  elevator,  the  bushel  expressed  in  cubic  feet  is  more  con- 
venient. A  cubic  foot  (1728  cu.  in.)  is  slightly  more  than 
.8  of  a  bushel  (2150.42  cu.  in.).  To  find  the  capacity  of  a 
bin,  then,  multiply  .8  bushel  by  the  number  of  cubic  feet  in 
the  bin.  For  greater  accuracy  add  |-  of  a  bushel  for  each 
100  cubic  feet.  If  capacity  in  heaped  bushels  is  desired, 
multiply  .63  bushels  by  the  cubic  feet  and  correct  as  above 
for  greater  accuracy. 

Thus,  a  bin  8  ft.  wide,  10  ft.  high,  and  20  ft.  long  would 
contain  1600  cu.  ft.,  and  .8  bu.  x  1600  =  1280  bu.  Adding 
■^  of  a  bushel,  the  contents  would  be  1285  bu. 

To  estimate  the  number  of  bushels  of  shelled  corn  in  a 
crib  of  ear  corn,  if  high  grade  and  dry,  take  J  as  many 
bushels  as  there  are  cubic  feet. 

Table 

There  are:      2  pints  (pt.)  in  1  quart  (qt.) 
8  quarts  in  1  peck  (pk.) 

4  pecks  in  I  bushel  (bu.) 

Note.  —  The  practical  worth  of  a  knowledge  of  dry  measure  consists 
not  alone  in  knowledge  of  its  units,  but  in  power  to  estimate  bulk  in 
terms  of  units.  Actual  measures  should  be  made  in  the  classroom,  and 
practice  given  in  estimating  bulk,  verifying  estimates. 

For  determining  the  number  of  bushels  of  different  grains  by  weights 
see  p.  139. 

87.  Liquid  Measure.  For  measuring  liquids,  the  standard 
unit  is  the  gallon.  It  contains  231  cu.  in.  A  cylindrical 
vessel  7  in.  in  diameter  and  6  in.  high  contains  a  standard 
gallon. 


MEASURES  OF  VOLUME  121 

For  finding  the  capacity  of  cisterns,  tanks,  etc.,  the  num- 
ber of  cubic  inches  may  be  divided  by  231  cu.  in.  Since  there 
are  1728  cu.  in.  in  a  cubic  foot,  there  would  be  7.4805"^  gallons 
in  a  cubic  foot.  A  shorter  method  for  finding  the  number 
of  gallons  in  a  tank  or  cistern  would  be,  then,  to  multiply  1^ 
gallons  by  the  number  of  cubic  feet,  and  subtract  -^^^  (^  of 
Y^)  of  itself.  This  will  give  the  correct  result  to  a  fraction 
of  a  gallon.  Thus,  a  cistern  6  x  8  x  15  ft.  would  contain 
720  cu.  ft.  7J  gal.  x  720  =  5400  gal.  ^  of  j^-^  of  5400 
gal.  =  J  of  54  gal.,  or  131  gal.  5400  gal.  -  131  gal.  =  53861 
gal. 

Casks,  hogsheads,  pipes,  tuns,  butts,  tierces,  carboys,  etc., 
are  indefinite  standards,  and  their  capacity  is  determined  by 
measurement,  and  it  is  usually  stamped  on  them.  Barrels  of 
oil  and.  liquors  are  also  indefinite.  The  barrel  of  31 J  gal. 
was  formerly  a  standard,  and  is  still  so  used  for  some  purposes. 

Table 


There  are:       4  gills  (gi)  in  1  pint  (pt.) 
2  pints  in  1  quart  (qt.) 

4  quarts        in  1  gallon  (gal.) 


88.  Apothecaries'  Liquid  Measure.  For  measuring  drugs, 
still  smaller  units  are  used.  The  gill  is  divided  into  fourths, 
known  as  fluid  ounces.  The  size  of  small  bottles  is  usually 
designated  by  the  number  of  fluid  ounces  they  contain.  The 
further  subdivisions  are  shown  in  the  following  table. 

Table 

There  are:    60  minims  (M)  in  1  fluid  dram 
8  fluid  drams  in  1  fluid  ounce 
16  fluid  ounces  in  1  pint 
8  pints  in  1  gallon 


122  ELEMENTS  OF  BUSINESS  ARITHMETIC 

PROBLEMS 

1.  How  many  pinfcs  in  6  quarts?    In  3  pecks?    In  4  bushels? 

2.  How  many  pints  in  8  quarts  ?    In  5  gallons  ?    In  6  gallons  and  3 
quarts  ? 

3.  How  many  quarts  in  5  pecks?  In  4  bushels?  In 3  bushels  3  pecks ? 
In  14  pecks  ? 

4.  What  will  8  bu.  10  qt.  cranberries  cost  at  12^  a  quart? 

5.  What  will  3  pk.  5  qt.  beans  costs  at  5)^^  a  quart? 

6.  Reduce  25  gal.  3  qt.  1  pt.  to  pints. 

7.  Change  196  pints  to  gallons. 

8.  Change  680  quarts  to  bushels. 

9.  Reduce  f  bu.  to  pints. 

10.  How  many  fluid  drams  in  3  fluid  ounces?  How  many  in  3  pints? 
How  many  in  1  gallon  ? 

11.  What  is  the  difference  in  the  size  in  cubic  inches  of  a  quart  dry 
measure  and  a  quart  liquid  measure  ? 

12.  A  bin  for  corn  is  8  x  9  x  24  ft.  How  many  bushels  of  shelled  corn 
can  be  obtained  from  the  ear  corn  it  would  hold  ? 

13.  A  swimming  pool  in  a  gymnasium  is  15  x  10  x  40  ft.  How  many 
gallons  of  water  will  it  hold  ? 

89.  Forms  of  Solids.  A  solid  is  a  body  of  matter  or  a 
defined  portion  of  space. 

A  solid,  whose  surfaces  are  planes  is  called  a  polyhedron. 
Polyhedrons  are  named  tetra,  hexa,  octa,  dodeca,  or  icosahe- 
drons,  as  they  have  four,  six,  eight,  twelve,  or  twenty  faces. 

A  prism  is  a  solid  whose  bases  are  equal,  similar,  and 
parallel,  and  whose  sides  are  parallelograms.  Prisms  are  de- 
scribed as  triangular,  square,  oblong,  etc.,  according  to  the 
form  of  their  bases.  They  are  right  or  oblique,  as  their 
sides  are  perpendicular  or  oblique  to  their  bases. 

A  solid  with  two  similar,  parallel,  and  plane  surfaces  as 
bases,  and  curved  surfaces  as  sides,  is  called  a  cylinder.  If 
the  bases  are  circles,  it  is  said  to  be  a  circular  cylinder.  In 
speaking  of  a  cylinder,  a  circular  cylinder  is  usually  meant. 


MEASURES  OF  VOLUME  123 

A  cone  is  a  solid  having  a  circle  for  a  base,  and  tapering 
to  a  point  or  vertex. 

A  sphere  is  a  solid  or  space  contained  within  a  given  sur- 
face, every  point  of  which  is  equidistant  from  a  central 
point  within.   • 

MEASUREMENT  OF  NON-RECTANGULAR  SOLIDS 

90.  Non-rectangular  Prisms.  In  studying  the  measure- 
ments of  area,  all  non-rectangular  forms  were  reduced  to 
equivalent  rectangles.  So  in  volume  all  solids  not  rectan- 
gular prisms  should  be  considered  as  equivalent  rectangular 
prisms,  and  then  the  volume  computed,  as  in  Sec.  78. 

In  all  non-rectangular  prisms,  first  determine  the  number 
of  cubic  units  in  one  layer  covering  the  base.  If  this  base 
were  rectangular,  there  could  be  placed  upon  it,  in  one  layer, 
as  many  cubic  units  as  there  are  square  units  in  the  area. 
By  finding  the  area,  then,  of  a  rectangle  equivalent  to  the 
base,  we  have  the  volume  of  one  layer.  The  altitude  or 
height  of  the  prism  is  the  number  of  layers. 

The  volume  of  one  layer  of  cubic  units  on  the  base  of  a  tri- 
angular^ trapezoidal^  or  any  other  non-rectangular  prism,  is 
equal  to  the  volume  of  one  layer  on  a  rectangle  equivalent  to  such 
hase^  and  there  are  as  many  such  layers  as  the  prism  is  corre- 
sponding units  high. 

91.  Volume  of  Pyramids.  Cut  from-  cardboard  four  tri- 
angles with  a  3-inch  base  and.5|-ihch  altitude.  Sew  the 
edges  together  to  form  a  pyramid,  having  an  altitude  of  5 
inches. 

Also  cut  four  rectangles  3  by  5  in.,  and  one  3  in.  square ; 
sew  the  edges  of  the  rectangles  together  and  to  the  edges  of 
the  square,  to  form  a  rectangular  prism  5  in.  long. 

The  prism  and  the  pyramid  thus  formed  have  the  same 
base,  3  in.  square,  and  their  common  altitude  is  5  in. 


124  ELEMENTS  OF  BUSINESS  ARITHMETIC 

Fill  the  pyramid  with  dry  sand,  and  empty  it  into  the 
prism.  It  will  require  three  times  the  contents  of  the  pyra- 
mid to  fill  the  prism.  Its  volume  is,  then,  one  third  that  of 
the  prism. 

The  volume  of  a  pyramid  is  equivalent  to  that  of  a  rectangular 
prism  with  the  same  base  arid  one  third  the  altitude. 

92.  Cylinders.  As  in  the  prisms  (Sec.  90),  the  number  of 
cubic  units  on  the  base  of  a  cylinder  would  be  the  same  as  on 
a  rectangle  equivalent,  to  the  area  of  the  base.  The  volume 
or  capacity  of  a  cylinder  is  equivalent  to  the  volume  of  a  rec- 
tangular prism  whose  base  is  one  half  the  circumference  in  length- 
and  the  radius  in  widths  and  whose  height  is  the  same  as  the 
length  of  the  cylinder. 

93.  Cones.  Make  from  pasteboard,  or  procure  a  hollow 
cone,  and  a  cylinder  having  the  same  base  and  altitude.  Fill 
tlie  cone  with  dry  sand  and  empty  it  into  the  cylinder. 
Three  times  the  contents  of  the  cone  will  be  found  necessary 
to  fill  the  cylinder. 

The  volume  of  a  cone  is  equivalent  to  one  third  that  of  a 
cylinder  with  the  same  base  and  altitude. 

94.  The  Sphere.  A  sphere  may  be  considered  as  made  up 
of  many  pyramids  having  their  bases  on  the  surface  and  their 
apexes  at  the  center.  Applying  the  principle  of  the  pyramid 
(Sec.  91),  the  following  statement  is  derived : 

The  volume  of  a  sphere  is  equivalent  to  that  of  a  rectangular 
prisjn  with  a  base  equal  to  the  surface  area  of  the  sphere  (Sec. 
76)  and  height  equal  to  one  third  its  radius. 

PROBLEMS 

1.  How  many  cubic  inches  in  a  prism  2  ft.  high,  the  sides  of  the  rec- 
tangular base  being  10  and  16  in.  respectively  ? 

2.  An  ash  hopper  is  in  the  form  of  a  pyramid.  If  the  base  is  6  ft. 
square  and  the  height  5  ft.,  how  many  bushels  of  ashes  will  it  hold? 


MEASURES  OF  VOLUME  125 

3.  A  cylindrical  cistern  is  12  ft.  in  diameter  and  15  ft.  deep.  How 
many  gallons  of  water  will  it  hold? 

4.  A  water  filter  in  the  form  of  a  cone  has  a  depth  of  10  ft.  The 
diameter  of  the  base  is  4  ft.     How  many  bushels  of  charcoal  will  it  hold? 

5.  The  diameter  of  a  cylindrical  tank  is  14  ft.,  the  length  is  18  ft. 
How  many  gallons  of  oil  will  it  hold  ? 

6.  A  man  dug  a  well  4  ft.  in  diameter  and  30  ft.  deep.  He  got  4^  a 
cubic  foot  and  the  dirt  for  digging.  He  sold  the  dirt  for  50^  a  cubic 
yard.     What  did  he  get  for  his  work  ? 

7.  A  man  has  a  bin  20  ft.  long  and  10  ft.  high.  If  it  is  4  ft.  wide 
at  the  bottom  and  6  ft.  at  the  top,  how  many  bushels  of  oats  will  it  hold  ? 

8.  How  many  cubic  feet  of  stone  in  a  pyramid  24  ft.  square  and  54 
ft.  high? 

9.  The  gas  tank  of  a  company  is  60  ft.  in  diameter,  and  when  full  of 
gas  stands  50  ft.  high.  How  many  cubic  feet  of  gas  has  been  used 
when  the  tank  stands  5  ft.  high  ? 

10.  Each  side  of  an  octagonal  room  is  12  ft.,  the  distance  from  the 
center  to  each  corner  is  16  ft.,  and  the  room  is  15  ft.  high.  How  many 
cubic  feet  of  air  does  it  contain  ? 

11.  A  well  is  34  ft.  deep  and  5  ft.  in  diameter.  The  w^ater  stands  10 
ft.  from  the  top.     How  many  standard  barrels  does  it  contain? 

12.  A  farmer  in  burying  his  potatoes  makes  3  cone-shaped  piles  3  ft. 
high  and  12  ft.  in  circumference  at  the  base.    How  many  bushels  has  he  ? 


IX 

MEASURES  OF  TIME 

95.  Unit.  The  unit  by  which  time  is  measured  is  the 
solar  day.  It  is  the  amount  of  time  required  for  the  earth 
to  make  one  complete  revolution  upon  its  axis.  It  is  com- 
puted from  the  instant  the  sun  is  at  the  highest  point  on  any 
stated  meridian  until  it  again  shines  at  the  highest  point  on 
that  meridian.  When  the  sun  is  at  the  highest  point,  it  is 
noon,  and  the  time  is  known  as  12  M.,  or  meridian.  If  the 
sun  has  passed  the  highest  point,  the  time  is  afternoon,  and 
it  is  marked  p.m.,  or  post-meridian  (after  meridian);  if  this 
time  has  not  yet  arrived,  the  time  is  forenoon,  and  it  is 
marked  a.m.,  or  ante-meridian  (before  meridian). 

The  day  as  used  in  practice  begins  when  the  sun  is  on  the 
opposite  side  of  the  earth,  or  to  midnight,  and  lasts  until 
the  succeeding  midnight. 

The  solar  year  is  the  time  required  for  the  earth  to  make 
one  complete  revolution  around  the  sun.  It  is  365  da.  5  hr. 
48  min.  49.7  sec.  long.  This  being  nearly  365^  da.,  the 
length  of  the  year  is  365  da.,  with  1  da.  added  in  each  4  yr., 
which  day  is  counted  as  the  29th  day  of  February.  Since 
in  4  yr.  the  5  hr.  48  min.  and  49.7  sec.  each  year  will  not 
amount  to  one  full  day,  the  extra  day  is  omitted  from  each 
centennial  year  not  divisible  by  400.  Thus,  the  year  1900 
was  not  a  leap  year,  but  the  year  2000  will  be. 

Note.  —  In  most  business  transactions  30  days  are  considered  1 
month,  and  12  months  a  year.  The  calendar  length  of  the  months  varies. 
January,  March,  May,  July,  August,  October,  and  December  have  31 
days ;  April,  June,  September,  and  November  have  30  days  ;  while  Feb- 
ruary has  28  days'  in  all  years  except  leap  years,  when  it  has  29  days. 

X26 


MEASURES  OF  TIME  127 

Table 
There  are :  60  seconds  (sec.)  in  1  minute  (min.) 


60  minutes 

in  1  hour      (hr.) 

24  hours 

in  1  day        (da.) 

7  days 

in  1  week     (wk.) 

30  days 

in  1  month  (mo.) 

12  months 

in  1  year      (yr.) 

365  days 

in  1  year 

366  days 

in  1  leap  year 

96.  Difference  in  Time.  There  are  two  methods  of  finding 
the  difference  in  time  between  certain  dates.  The  exact 
method  takes  into  account  the  calendar  months  intervening. 
It  is  used  in  determining  the  date  of  maturity  of  notes  or 
other  legal  paper  when  the  time  period  is  given  in  days,  and 
also  by  many  banks  in  calculating  discount.  The  eorhmercial 
method  counts  the  year  as  made  up  of  12  months  of  30  days 
each.  In  this  method  the  difference  in  time  is  usually 
found  by  what  is  termed  compound  subtraction, 

97.  Compound  Subtraction  in  Time.     If  we  desire  to  find 

the  difference  in  time  between  May  24, 1918,  and  March  16, 

1922,  we  write  the  dates  as  above,  ^  ^^^  «  -m  j 

,          -              ,      ,  1922  3  mo.  16  da. 

puttinsr  months  under  months,  days  ^rv^D  c  ^a  j 

-.1                    .     _  .  _           -^  1918  5  mo.  24  da. 

under  days,  etc.     As  24  days  can-  5 7: ^7^7-3 — 

1         ,.          -.  /»    1           1          o  yr.  9  mo.  22  da. 
not    be    taken   from   16   days,    1 

month  must  be  reduced  to  days,  making  46  days  in  all. 
There  would,  then,  be  a  difference  of  22  days.  One  month 
being  used,  there  are  but  2  months  remaining  in  the  minu- 
end, and  1  year  or  12  months  must  be  added,  making  14 
months.  Subtracting  5  months,  the  remainder  is  9  months. 
One  year  of  1922  being  used,  the  date  becomes  1921;  and 
subtracting  1918,  we  have  3  years.  The  whole  difference 
in  time,  then,  is  3  years,  9  months,  22  days. 


128  ELEMENTS  OF  BUSINESS  ARITHMETIC 

LONGITUDE  AND  TIME 

98.  Measurement  of  Circles.  For  measuring  angles,  a  unit 
called  a  degree  is  used.  It  is  -^^  of  a  right  angle.  The  en- 
tire angular  measure  around  the  center  of  a  circle  is  equiva- 
lent to  four  right  angles,  or  360  degrees. 

The  part  of  a  circumference  opposite  each  degree  angle, 
or  g^Q  of  a  circumference,  is  also  called  a  degree  (written  °). 
In  the  same  or  equal  circles,  one  degree  of  arc  measures  one 
degree  of  angle.  If  the  circle  is  larger,  the  length  of  the 
arc  increases,  while  the  angle  remains  the  same.  The  degree 
of  arc  means,  simply,  ^l^  of  a  circumference.  The  derived 
units  are  shown  in  the  table  given  below. 

Table 

There  are :    60  seconds  ('^)  in  1  minute  (') 
60  minutes         in  1  degree  (°) 
360  degrees         in  1  circle 

99.  Latitude  and  Longitude.  When  the  units  of  circular 
measure  are  applied  to  the  measurement  of  the  earth's  circum- 
ference, they  are  of  use  for  locating  places  on  the  globe  and 
for  measuring  distances. 

For  fixing  locations  north  and  south,  great  imaginary  cir- 
cles termed  parallels  are  drawn  one  degree  apart  and  parallel 
with  the  equator.  The  number  of  degrees  north  or  south 
of  the  equator  determines  the  latitude.  These  circles  being 
parallel,  the  distance  between  them,  or  the  length  of  a  degree 
of  latitude,  is  always  nearly  the  same.  Owing,  however,  to 
the  curvature  of  the  earth,  a  degree  of  latitude  varies  slightly, 
it  being  68.72  miles  at  the  equator,  and  69.34  miles  at  the 
poles.  Sixty-nine  and  sixteen  hundredths  miles  has  been 
adopted  as  the  standard  length  of  a  degree  of  latitude.  -^^  of 
a  degree  of  latitude  is  sometimes  known  as  a  geographical 


MEASURES  OF  TIME  129 

mile,  to  distinguish  it  from  the  ordinary  or  statute  mile. 
The  geographical  mile  is,  therefore,  equal  to  l.lo2|  statute 
miles. 

For  measuring  east  and  west,  great  circles,  called  me- 
ridians, pass  through  the  poles.  The  distance  between  them 
varies  from  nothing  at  the  poles,  where  they  meet,  to  ^1^  of 
the  circumference  of  the  earth  at  the  equator.  A  degree  of 
longitude  varies  in  length,  then,  from  69.16  miles  at  the 
equator  to  nothing  at  the  poles. 

Distance  east  or  west  is  indicated  by  the  number  of 
degrees  or  meridians  east  or  west  of  a  given  meridian  desig- 
nated as  a  principal  meridian  for  such  purpose.  The  me- 
ridians of  Greenwich,  England,  and  Washington,  D.C.,  are 
both  used  as  principal  meridians. 

100.    Difference  in  Longitude.     If  two  places  are  both  east 
or  both  west  of  the  principal  meridian,  the  difference  in  their 
longitude  is  found  by  subtracting  the 
longitude  of  one  from  that  of  the  otjier.  /-^n 

Thus,  the  longitude  of  New  York  is 
74°  0'  3''  W.  (Greenwich),  and  that  of  ^'  "^^  ^"^  ^* 
Chicago  is  87°  36^  ^2"  W.  To  find  ^4  0  3  W. 
the  difference,  write  the  less  number 
under  the  greater,  putting  degrees 
under  degrees,  minutes  under  minutes, 
and  seconds  under  seconds  (<x).  When 
the  minuend  numbers  are  larger,  ordi- 
nary subtraction  will  give  the  result. 
In  this  case  it  is  13°  36'  39'^  Should 
some  of  the  minuend  numbers  be 
smaller,  the  same  method  as  that  used 
in  finding  the  difference  of  time  (Sec. 
97)  should  be  used.  Thus,  in  finding 
the  difference  in  longitude  between  New      76°     20'     18'' 


13° 

36'     39" 

(J) 

84° 

26'       0" 

74° 

0'       3" 

10° 

25'     57" 

(0 

74° 

0'       3"  W. 

2° 

20'     15"    E. 

Cd)          , 

87° 

36'     42''  W. 

13° 

23'     43"    E. 

130  ELEMENTS  OF  BUSINESS  ARITHMETIC 

York  and  Cincinnati  (84°  26'  0'  W.) 
(6),  it  is  necessary  to  change  one  minute 
to  seconds  before  the  subtraction  of  3" 
can  be  made.  101°       0^     25" 

If  the  two  places  are  one  east  and  one 
west  of  the  principal  meridian,  the  difference  in  their  longi- 
tude is  found  by  adding  them.  Thus,  the  difference  in 
longitude  between  New  York  and  Paris  (2°  20'  15"  E.)  is 
found,  by  adding,  to  be  76°  20'  18"  ((?),  and  between  Chicago 
and  Berlin  (13°  23'  43"  E.)  is  found  to  be  101°  0'  25"  {d). 

101.  Relative  Time.  Difference  in  longitude  is  often 
expressed  by  the  difference  in  the  time  of  two  given  places. 
When  the  time  of  two  places  is  given  at  the  same  instant, 
their  difference  in  time  may  be  found  by  compound  subtrac- 
tion. If  the  times  of  two  places  are  both  A.M.  or  both  P.M., 
the  difference  in  their  time  is  found  by  subtracting  (a). 


C«) 

New  York  4  hr. 

55  min. 

53    sec.  P.M. 

Boston         4  hr. 

44  min. 

15    sec.  P.M. 

11  min.     38    sec. 

When  the  times  given  are  one  a.m.  and  one  p.m.,  the  dif- 
ference in  their  time  is  found  by  adding  12  hr.  to  the  p.m. 
time  and  subtracting  the  a.m.  time  from  the  sum  (5). 


(^) 

Berlin 

Ihr. 

20  min. 

42    sec.  P.M. 

Chicago 

6hr. 

36  min. 

40J  sec.  A.M. 

6  hr.     44  min.       1|  sec. 

102.  Comparative  Tables.  As  the  earth  revolves  through 
360°  of  longitude  in  24  hours,  it  revolves  through  1°  in  ^J-j^ 
of  24  hours,  or  4  minutes.  In  this  way  the  following  tables 
are  derived. 


MEASURES  OF  TIME 


131 


Table  1 
There  are  :   360°  longitude  in  24  hours  of  time 

1°  longitude  in  4  minutes  of  time 
1'  longitude  in  4  seconds  of  time 
1"  longitude  in  -f^  seconds  of  time 

Table  2 
There  are :  24  hours  for  360° 

1  hour  for  15° 
1  minute  for  15' 
1  second  for  15" 

103.  Reduction.  Inasmuch  as  the  time  of  day  depends 
upon  the  position  of  the  sun  in  the  heavens,  a  place  west 
of  another  will  have  an  earlier  time,  and  a  place  east  will 
have  a  later  time.  A  difference  of  longitude,  then,  will 
make  a  difference  in  time,  and  if  we  know  the  difference  in 
longitude  between  two  places,  we  may  calculate  their  differ- 
ence in  time.  Thus,  the  difference  in  longitude  between 
New  York  and  Paris  is  76°  20'  18''.  Since  1°  in  longitude 
makes  a  difference  of  4  minutes  in  time,  76°  would  make  a 
difference  of  304  minutes  (4  min.  x  76),  or  5  hr.  4  min.  1' 
making  4  sec.  difference,  20.3'  (20'  18")  would  make  81.2 
sec,  or  1  min.  21.2  sec.  All  together,  there  would  be  a  dif- 
ference in  time  of  5  hr.  5  min.  21.2  sec.  The  problem  may 
be  written  in  the  manner  shown, 
and  solved  by  the  method  of 
compound  multiplication,  which 
combines  multiplication  of  each 
part  and  a  reduction  to  higher 
denominations. 

The  first  product  (a)  is  that  from  simple  multiplication, 
and  the  second  (5)  is  the  result  of  the  reduction.  The 
seconds  of  longitude  should  be  reduced  to  decimals  of  a 
minute  before  multiplication. 


76 


20.3 
4 


(a)  304     81.2 

(5)  5  hr.  5  min.  21.2  sec. 


132  ELEMENTS  OF  BUSINESS  ARITHMETIC 

On  the  other  hand,  when  the  difference  in  time  between 

two   places   is   known,    the   difference    ,^c^r^r      •      r»-i  n 

.     ,        .,    T  1      n        11       T   .1     4)305  mm.  21.2  sec. 

in  longitude  may  be  lound  by  divid-         r-^o orT^r 

ing  such  time  by  4,  using  the  method 

of  compound  division.  Thus,  if  the  difference  in  time  be- 
tween New  York  and  Paris  were  known  to  be  5  hr.  5  min. 
21.2  sec.  (305  min.  21.2  sec),  the  longitude  difference 
would  be  found  by  taking  |  of  the  time  (expressed  in  min- 
utes and  seconds).^     This  would  be  76°  20.3'  (T6°  20'  18''). 

The  minute  remaining  in  the  first  division  is  changed  to 
seconds  and  added  to  the  second  dividend  before  again 
dividing  by  four. 

104.  Standard  Time.  If  15°  of  longitude  make  a  differ- 
ence of  one  hour  in  time  (Sec.  102),  it  is  plain  that  as  one 
traveled  west  the  time,  as  shown  by  his  watch,  would  grow 
faster  than  the  local  (meridian)  time  along  the  route.  This 
varying  time  led  to  many  inconveniences,  particularly  in 
railroad  management. 

In  1883,  a  system  of  standard  time  was  adopted,  and  it 
is  now  in  general  use.  By  this  plan,  the  United  States  is 
divided  into  four  time  belts,  each  about  15°  in  width. 

Over  the  whole  of  each  belt  exactly  the  same  time  is  used. 
It  is  the  local  or  sun  time  at  each  standard  meridian.  These 
meridians  are  the  75th,  which  passes  near  Philadelphia ;  the 
90th,  near  St.  Louis  ;  the  105th,  near  Virginia  City,  Mont. ; 
and  the  120th,  near  Carson  City,  Nev.  The  belts  are  known 
respectively  as  Eastern,  Central,  Mountain,  and  Pacific,  and 
extend  about  7J°  on  either  side  of  the  meridian.  The  stand- 
ard meridians,  being  15°  apart,  the  difference  in  time  between 
them  and  their  belts  is  exactly  one  hour.  Hence,  in  travel- 
ing westward,  watches  should  be  set  back  one  hour  on  cross- 
ing the  dividing  line  between  these  belts,  and  in  traveling 
eastward  set  forward  one  hour. 


MEASURES  OF  TIME 


133 


The  dividing  line  between  the  belts  is  quite  irregular, 
owing  to  the  necessity  of  railroads  avoiding  a  change  of 
time  at  any  but  division  stations.     Similar  devices  for  using 


BXiMDABD  XIM£  IS  ICHE  IFMl.TF.n  aiiXES 


exactly  the  same  time  over  given  areas  are  now  in  use  in 
the  countries  of  Europe  and  in  the  European  colonies  of 
Asiatic  and  East  Indian  countries. 

105.  International  Date  Line.  In  traveling  around  the 
earth  east  or  west,  the  fact  that  the  earth  is  rotating  from 
west  to  east  has  an  effect  upon  the  apparent  time  of  day. 
In  traveling  east,  the  earth  travels  with  one,  but  the  sun 
seems  to  travel  in  an  opposite  direction,  and  one  therefore 
shortens  his  day.  In  traveling  around  the  earth  to  the  east, 
one's  time  would  be  so  shortened  that  it  would  take  a  day 
more  for  the  journey  than  the  actual  time  consumed  would 
make  if  divided  into  days  of  twenty-four  hours  each. 

On  the  other  hand,  if  one  traveled  west  around  the  world, 
it  would  seem  to  take  a  day  less  than  the  actual  time  con- 
sumed. 

In  navigation,  this  phenomenon  caused  annoyance,  for  ships 
going  in  opposite  directions  would  often  be  a  day  apart  in 


134 


ELEMENTS  OF  BUSINESS  ARITHMETIC 


their  dates.  To  avoid  this,  a  line  called  "  The  International 
Date  Line  "  has  been  established.  This  line  follows,  approxi- 
mately, the  180th  meridian  from  Greenwich,  diverging  from 
such  meridian  whenever  necessary  to  avoid  land,  because  of 
the  confusion  of  dates  that  would  result  through  the  use  of 
different  dates  on  the  same  body  of  land.  The  fact  that 
the  180th  meridian  east  or  west  of  Greenwich  passes  between 


105°        136° 


e)l0lGlQlQlel©l0lOlQ 


000 

3  P.M.      5  P.M.      7  pIm. 


V\\' 


5 


EB, 


8  PIM. 


EaBt  180°         160°         120°  90°  60° Long.  30°  West  0°  Long.  30°  Eaat  60°  90°  120' 

iNTERNATIONAIi  DATE  LiNE 

North  America  and  Asia,  and  traverses  the  ocean  save  for 
a  few  small  islands,  makes  it  a  convenient  longitude  for  such 
a  purpose.  Whenever  ships  pass  this  line  going  eastward, 
they  subtract  a  day  from  their  reckoning,  and  when  going 
westward,  they  add  a  day. 

106.  Longitude  of  Leading  Cities.  The  longitude  of  the 
cities  given  below  may  be  determined  from  their  difference 
in  time  from  the  time  of  Greenwich,  which  is  given  with  the 
direction  indicated. 

H.     M.         S.  H.    M.         S. 

Albany  4    55      6.8    W.    Annapolis,  Md.     5      5    56.5    W. 

Ann  Arbor,  Mich.  5    34    55.2     W.    Baltimore,  Md.     5      6    26.0    W. 


DIFFERENCE  IN  TIME  FROM  GREENWICH        135 


H. 

M. 

8. 

H. 

M. 

s. 

Berlin 

0 

53 

34.9 

E. 

Hong  Kong 

7 

36 

41.9 

E. 

Bombay 

4 

51 

15.7 

E. 

Liverpool 

0 

12 

17.3 

W. 

Bordeaux,  France 

0 

2 

5.4 

W. 

Madrid,  Spain 

0 

14 

45.4 

W. 

Brussels,  Bel. 

0 

17 

28.6 

E. 

Melbourne,  Vic. 

9 

39 

54.1 

E. 

Calcutta 

5 

53 

20.7 

E. 

Moscow 

2 

30 

17.2 

E. 

Cambridge,  Eng. 

0 

0 

22.7 

E. 

New  Orleans 

6 

0 

13.9 

W, 

Canton,  China 

7 

33 

46.3 

E. 

New  York 

4 

55 

54.6 

W. 

Cape  Good  Hope 

1 

13 

58.0 

E. 

Paris 

0 

9 

20.9 

E. 

Chicago 

5 

50 

26.7 

W. 

Philadelphia 

5 

0 

38.5 

W. 

Cincinnati 

5 

37 

41.3 

W. 

Rome 

0 

49 

55.6 

E. 

Denver,  Col. 

6 

59 

47.6 

W. 

San  Francisco 

9 

9 

42.8 

W. 

Dublin,  Ireland 

0 

25 

21.1 

W. 

Seattle 

8 

9 

19.9 

W. 

Edinburgh 

0 

12 

43.1 

W. 

St.  Louis 

6 

0 

49.1 

W. 

Glasgow 

0 

17 

10.6 

W. 

Providence,  R.I. 

4 

45 

37.5 

W. 

PROBLEMS 

1.  A  note  dated  May  15  was  paid  July  6.    How  many  days  did  it 
run?  ^ 

Solution  :  16  days  in  May. 

30  days  in  June. 
_6  days  in  July. 
52  days  from  May  16  to  July  6. 

Note.' —  The  day  on  which  the  note  is  given  is  not  counted,  but  the 
day  on  which  it  is  due  is  counted. 

Find  exact  number  of  days  from : 

2.  Aug.  7  to  Oct.  28. 

3.  Dec.  14  to  Feb.  12. 

4.  May  20  to  July  25. 

5.  Aug.  18  to  Nov.  14. 

6.  Sept.  28  to  Jan.  31. 

By  compound  subtraction  find  difference  in  time  between : 

1.  Nov.  18,  1903,  July  7,  1906.  4.   Oct.  17,  1903,  Dec.  6,  1907. 

2.  May  17,  1899,  Sept.  16,  1903.  5.   Aug.  26,  1902,  Feb.  15,  1908. 

3.  Feb.  11,  1910,  Jan.  17, 1903.  6.  May  25,  1901,  Aug.  3, 1904. 


7.  March  13  to  Nov.  26. 

8.  May  3  to  Dec.  5. 

9.  Jan.  1  to  March  13. 
10.  Sept.  15  to  Dec.  31. 


136  ELEMENTS  OF  BUSINESS  ARITHMETIC 

Reduction  : 

1.  The  difference  in  time  between  two  places  is  2  hr.  15  min.  What 
is  the  difference  in  longitude  ? 

2.  When  it  is  noon  at  Chicago,  what  time  is  it  15°  15'  west  of  Chi- 
cago?   32°  15' 30"  east? 

3.  A  man  travels  from  Cincinnati  until  his  watch  is  45  minutes  fast. 
In  what  direction  and  through  how  many  degrees  has  he  traveled? 

4.  What  is  the  difference  in  time  between  two  places  whose  longi- 
tudes are  80°  W.  and  65°  W.  ?    Two  places  16°  18'  E.  and  90°  0'  12"  W.  ? 

5.  By  use  of  the  preceding  table,  find  the  longitude  of  Albany,  Balti- 
more, Berlin,  Calcutta,  Dublin,  Madrid,  Paris,  Rome,  and  Providence. 

6.  When  it  is  11  a.m.  at  Washington,  it  is  10  hr.  7  min.  4  sec.  at  St. 
Louis.     What  longitude  is  St.  Louis  west  of  Washington? 

7.  Determine  the  time  and  date  at  Washington,  D.C.,  Hongkong, 
Cape  Good  Hope,  and  Moscow  when  it  is  midnight,  Jan.  1,  at  Green- 
wich. 

8.  When  it  is  noon,  Feb.  15,  at  Glasgow,  what  time  and  date  is  it  at 
Denver  ?    At  Melbourne  ?    At  New  York  ?    At  Brussels  ? 

9.  When  it  is  8  p.m.  at  Philadelphia,  what  is  the  time  at  Liverpool? 
At  Paris?    At  Berlin?    At  Rome? 


MEASURES  OF  WEIGHT 

107.  Units  of  Weight.  Weight  is  the  downward  pressure 
of  bodies  toward  the  earth,  or  the  measure  of  the  attraction 
of  gravitation.  The  English  Ttoj  pound  has  been  adopted 
as  the  unit.  As  fixed  by  the  English  law,  this  is  the  weight 
of  22.7944  cu.  in.  of  pure  water  at  its  greatest  density. 
The  use  of  this  unit  and  its  Troy  subdivisions  is  restricted 
to  weighing  gold,  silver,  jewels,  etc.  The  Troy  pound  is 
divided  into  twelfths,  known  as  ounces,  which  in  turn  are 
divided  into  20  pennyweights,  each  composed  of  24  grains. 
There  are,  then,  5760  grains  in  a  pound  Troy. 

Troy  Weight  Table 
There  are : 

24  grains  (gr.)     in  1  pennyweight  (pwt.) 
20  pennyweights  in  1  ounce  (oz.) 
12  ounces  in  1  pound  (lb.) 

Comparisons 

1  lb.  =  12  oz.  =  240  pwt.  =  5760  gr. 

1  oz.  =    20  pwt.  =    480  gr. 

1  pwt.  =      24  gr. 

108.  Apothecaries'  Weight.  For  weighing  drugs,  in  com- 
pounding prescriptions,  etc.,  the  Troy  ounce  is  differently 
divided.     The  table  of  Apothecaries'  weight  follows. 

137 


138  ELEMENTS  OF  BUSINESS  ARITHMETIC 

Table 


There  are : 


TABLE 

20  grains  (gr.)  in  1  scruple  (sc.) 
3  scruples         in  1  dram  (dr.) 
8  drams  in  1  ounce  (oz.) 

12  ounces  in  1  pound  (lb.) 

109.  Avoirdupois  Weight.  For  weighing  ordinary  bulky- 
articles,  such  as  coal,  grain,  groceries,  etc.,  the  Avoirdupois 
pound  is  used.  It  consists  of  7000  Troy  grains,  and  has 
been  fixed  by  Congress  (1901)  as  the  weight  of  27.7015  cu. 
in.  of  distilled  water,  at  62°  Fahrenheit,  weighed  with  brass 
weights  in  air,  with  the  barometer  at  30  in. 

Table 

There  are  :    16  ounces  (oz.)       in  1  pound  (lb.) 

100  pounds  in  1  hundredweight  (cwt.) 

20  hundredweight  in  1  ton  (T.) 

Comparisons 

1  ton  =  2000  lb. 
1  lb.  =  7000  grains 
1  oz.  =  437.5  grains 

Note.  —  At  the  Custom  House,  and  to  some  extent  in  mining,  the 
long  ton  of  2240  pounds  is  still  used.  As  a  subdivision,  112  pounds  are 
called  a  hundredweight. 

110.  Bushel  Weights.  In  handling  grains  and  many 
other  products  for  shipment,  the  dry-measure  units  are  sel- 
dom used.  It  is  much  more  convenient  to  weigh  in  bulk 
and  allow  a  certain  weight  per  bushel.  Many  states  have, 
by  statute,  fixed  the  number  of  pounds  per  bushel  of  various 
products.  The  result  of  this  practice  has  been  a  lack  of 
uniformity  among  the  states.  In  1901  Congress  fixed  t^ 
minimum  weight  per  bushel  of  certain  articles  of  produce 


MEASURES  OF  WEIGHT-  139 

Pounds  Pounds 

Wheat 60  Dried  apples 26 

Corn,  ear 70  Dried  peaches 33 

Corn,  shelled 56  Clover  vseed 60 

Buckwheat 48  Flax  seed 56 

Barley 48  Millet  seed 50 

Oats 32  Hungarian 50 

Peas 60  Timothy  seed 45 

White  beans 60  Blue  grass 44 

Castor  beans 46  Hemp  seed 44 

Rye 56  Corn  meal 48 

White  potatoes 60  Ground  peas 24 

Sweet  potatoes 55  Bran .20 

Onions 57  Malt 34 

Turnips 55 

Note. — In  many  places  the  weight  per  bushel  has  been  discarded, 
grain  and  other  materials  being  bought  by  hundredweight.  This  plan 
has  many  advantages. 

Salt.  The  weight  per  bushel  of  salt  as  adopted  by  dif- 
ferent states  ranges  from  50  to  80  pounds.  Coarse  salt  in 
Pennsylvania  is  reckoned  at  TO  pounds,  and  in  Illinois  at 
50  pounds  per  bushel.  Fine  salt  in  Pennsylvania  is  reck- 
oned at  62  pounds,  in  Kentucky  and  Illinois  at  55  pounds 
per  bushel. 

PROBLEMS 

1.  How  many  pounds  in  1  T.  6  cwt.  ?    In  4  cwt.  ?    In  6  T.  3  cwt.  ? 

2.  How  many  ounces  in  5  lb.?  In  8  lb.?  In  10  lb.?  In  16  lb.? 
(Troy.) 

3.  How  many  grains  in  3  pwt.?  In  5  pwt.?  In  8  pwt.?  In 
12  pwt.?    In  16  pwt.? 

4.  How  many  tons  in  40  cwt.?  In  50  cwt.?  In  60  cwt.?  In  80 
cwt.?    In  100  cwt.? 

5.  How  many  ounces  Troy  in  4  lb.  ?    In  8  lb.  ?    In  20  lb.  ?    In  15  lb.  ? 

6.  How  many  grains  in  3  oz.  16  pwt.  12  gr.  ?    In  4  oz.  6  pwt.  20  gr.  ? 


140  ELEMENTS  OF  BUSINESS  ARITHMETIC 

7.  How  many  bushels  of  ear  corn  in  a  load  weighing  2650  lb.? 

8.  What  will  2680  lb.  shelled  corn  cost  at  60^  per  bushel? 

.15      1430 
Solution  :  $.^0  x  %^^^  -  %  30.64: 

u 

7 

9.   What  will  3670  lb.  wheat  cost  at  78  f  per  bushel  ? 

10.  What  will  2750  lb.  potatoes  cost  at  45  ^  per  bushel  ? 

11.  If  a  man  weighs  156  lb.  Avoirdupois,  what  will  he  weigh  in  Troy- 
weight  ? 

12.  A  farmer  raises  7600  lb.  rye,  1856  lb.  clover  seed,  1375  lb.  flax, 
4532  lb.  onions,  458  lb.  timothy  seed.  How  many  bushels  of  each  did  he 
raise  ? 

13.  A  cubic  foot  of  water  weighs  62|  lb.  At  that  rate,  what  will  be 
the  weight  of  the  water  in  a  tank  4  x  6  x  15  ft.  ? 

14.  A  grocer's  boy  sold  10  lb.  coffee  by  mistake,  weighing  it  on  a 
druggist's  scales.  How  much  did  the  grocer  lose  or  gain,  if  the  coffee 
was  worth  20  ^  a  pound  ? 

15.  A  farmer  sold  27,650  lb.  corn  at  65  ^  per  bushel,  48,500  lb.  wheat 
at  82/  per  bushel,  15,810  lb.  oats  at  35/  per  bushel.  Find  the  total 
selling  price. 

16.  A  horse  requires  12  lb.  of  oats  and  20  lb.  of  hay  per  day.  What 
will  it  cost  to  feed  him  a  year,  if  oats  are  worth  26/  a  bushel,  and  hay 
$  9.75  a  ton  ? 

17.  A  dealer  purchased  in  a  day  26,040  lb.  oats  at  28/  a  bushel, 
19,654  lb.  wheat  at  76  /  per  bushel,  28,642  lb.  corn  at  49  /  per  bushel,  and 
15,060  lb.  barley  at  98  /  per  bushel.  The  oats  are  sold  at  an  advance  of 
4  /,  the  wheat  at  5  /,  the  corn  at  7  /,  and  the  barley  at  6  /.     Find  the  gain. 

18.  A  farmer  raised  on  his  farm  28,540  lb.  wheat,  27,486  lb.  rye, 
8752  lb.  potatoes,  54,660  lb.  hay,  40,500  lb.  oats,  and  600  lb.  beans.  He 
sold  the  wheat  for  65  /,  the  rye  for  57  /,  the  potatoes  for  $  1.25,  the  beans 
for  $  3.50,  the  oats  for  32  /  per  bushel,  and  the  hay  for  $  8.75  per  ton. 
Find  total  returns.  Find  his  gain,  if  he  paid  %  375  for  labor,  $  245  for 
seed,  $175  for  machinery,  and  $60  for  hauling  the  crops  to  market. 

19.  A  coal  dealer  bought  326  tons  of  coal  by  the  long  ton  at  $  3.75 
per  ton,  and  sold  it  by  the  short  ton  at  $6.50.     How  much  did  he  gain  ? 


XI 

MEASURES  OF  VALUE 

111.  The  Unit  of  Value.  Congress  has  designated  the  dol- 
lar as  the  unit  of  value  for  the  United  States,  to  be  com- 
posed of  23.22  grains  of  pure  gold.  For  greater  hardness, 
the  pure  metal  is  mixed  with  copper  or  an  alloy  of  copper 
and  silver,  nine  parts  gold  and  one  of  alloy,  making  the  full 
weight  of  a  gold  dollar  25.8  grains.  The  silver  dollar  con- 
tains 371.25  grains  of  pure  silver;  41.25  grains  of  copper 
are  added  for  hardness. 

From  the  standard  unit,  f  1,  other  units  are  derived,  using 
decimal  divisions  and  multiples;  thus,  810  is  called  an  eagle, 
■^^  of  f  1  is  called  a  dime  (French  disme  or  ten)  or  ten  cents  ; 
^^^  of  a  dollar  is  1  cent  (Latin  centum  or  a  hundred),  and 
l-Q^^  of  a  dollar  is  1  mill  (Latin  mille  or  a  thousand).  While 
theoretically  all  these  divisions  are  made,  practically  we  use  but 
dollars  and  cents  in  writing  or  reading  expressions  of  value. 

112.  Coins.  For  the  convenience  of  business,  Congress 
has  caused  to  be  minted  the  following  coins : 

Grold.  Twenty  dollars,  sometimes  called  the  double  eagle, 
ten  dollars  or  eagle,  and  five  dollars  or  half  eagle. 

Silver,  One  dollar,  half  dollar  or  fifty  cents,  quarter  or 
twenty-five  cents,  and  dime.  All  except  the  first  are  subsid- 
iary. 

Minor  Coins,     Nickel  or  five  cents,  and  one-cent  piece. 

Note.  —  The  coinage  of  the  gold  dollar  was  discontinued  pursuant  to 
an  act  of  Congress,  Sept.  26,  1890. 

113.  Paper  Money.  For  larger  denominations,  and  for 
convenience  in  handling  larger  amounts  of  money.  Congress 

141 


142  ELEMENTS  OF  BUSINESS  ARITHMETIC 

has  authorized  the  issue  of  five  different  kinds  of  paper 
money,  printed  in  various  denominations.  These  are  United 
States  Notes  or  Greenbacks,  Silver  Certificates,  Gold  Certif- 
icates, Treasury  Notes  of  1890,  and  National  Bank  Notes. 

114.  Legal  Tender.  To  facilitate  trade  and  render  it 
more  secure,  governments  declare  certain  kinds  of  money 
to  be  "legal  tender,"  i.e.  money  that  must  be  accepted  by  a 
creditor  in  payment  of  a  debt. 

The  following  kinds  of  money  are  full  legal  tender :  all 
gold  coins,  silver  dollars.  Treasury  notes  of  1890,  and  United 
States  notes  (except  for  duties  on  imports  and  interest  on  the 
public  debt).  National  Bank  notes  and  gold  and  silver  certif- 
icates are  not  legal  tender,  but,  because  they  are  redeemable 
at  the  United  States  Treasury  in  legal  tender  money,  they  pass 
current  without  question.  Subsidiary  silver  coins  are  legal 
tender  for  ten  dollars  or  less,  and  minor  coins  for  25  ^  or  less. 

115.  Sending  Money  and  Valuables.  For  a  fee  of  10  cents 
a  letter  or  parcel  containing  money  or  other  articles  of  value 
may  be  sent  by  registered  mail.  Registered  mail  is  delivered 
only  to  the  addressee,  or  on  his  written  order,  and,  if  not 
known  to  the  authorities,  such  addressee  must  be  identified. 
A  receipt  is  taken  upon  delivery,  and  is  returned  to  the 
sender  of  the  letter  or  parcel.  This  receipt  is,  under  the  law, 
prima  facie  evidence  of  delivery.  In  case  of  loss,  the  gov- 
ernment indemnifies  the  sender  for  a  value  up  to  $50. 

B^  Express.  Express  companies  will  transport  money  under 
guarantee  of  its  safe  delivery,  and  will  indemnify  losses  in 
full.  The  cost  of  such  service,  however,  and  its  inconvenience, 
render  it  little  used  for  ordinary  business  transactions. 

Exchange.  By  exchange  is  meant  a  system  of  making 
money  payments  without  the  actual  transportation  of  money. 
It  is  accomplished  through  the  exchange  of  credits.  The 
principal  systems  of  exchange  are  given  below. 


MEASURES  OF  VALUE  143 

116.  Post  OflBice  Money  Orders.  The  government  provides 
a  simple  and  safe  means  of  exchange  through  the  postal  sys- 
tem. Offices  designated  as  "money-order  offices"  are  em- 
powered to  issue  orders  on  other  similarly  designated  offices 
for  an  amount  not  exceeding  f  100  to  any  one  order.  For 
sums  above  $100  more  than  one  order  must  be  purchased. 

To  obtain  a  money  order,  a  prescribed  form  of  application 
blank  must  be  filled  out.  The  rates  charged,  in  addition  to 
the  face  value  of  the  order,  are  printed  on  the  back  of  the 
order  blank,  and  the  cost  of  an  order  for  a  given  amount 
may  be  easily  computed.  The  government  is  liable  for  pay- 
ment to  the  wrong  person,  provided  the  wrong  payment  was 
not  brought  about  through  the  fault  of  the  remitter,  payee, 
or  indorsee.  Money  orders  may  be  obtained  or  cashed  at 
over  35,000  offices  in  the  United  States. 

117.  Express  Money  Orders.  Express  companies  issue 
money  orders  in  much  the  same  form  and  at  practically  the 
same  rates  as  the  Post  Office  Department.  A  receipt  is 
given  the  sender,  and,  in  case  the  order  is  lost,  a  refund  may 
be  had  upon  filing  an  indemnity  bond  guaranteeing  the  com- 
pany against  loss  should  the  lost  order  be  cashed. 

The  limit  of  each  order  is  $  50,  but  any  number  may  be  pur- 
chased. When  indorsed,  these  orders  are  usually  accepted 
by  banks  at  their  face  value. 

118.  Bank  Exchange.  Banking  houses  maintain  deposits 
or  accounts  with  other  banks,  called  "correspondents." 
They  sell  orders,  called  drafts,  on  these  correspondents, 
charging  a  small  fee  therefor.  In  all  ordinary  business  and 
for  relatively  small  sums,  the  fee  is  a  fixed  charge.  Ten 
cents  for  small  amounts,  and  fifteen  cents  per  hundred  dollars, 
are  the  rates  charged  in  many  banks.  For  larger  amounts, 
and  between  large  business  houses,  a  certain  fraction  (per 
cent)  of  the  amount  of  the  draft  issued  is  sometimes  charged. 


144  ELEMENTS  OF  BUSINESS  ARITHMETIC 

Note.  —  As  this  phase  of  exchange  deals  with  per  cents  and  banking 
procedure,  it  is  reserved  for  further  discussion  in  the  chapter  on  "  Bank- 
ing and  Discount." 

Some  banks  issue  drafts  payable  at  any  one  of  a  number  of 
other  banks.  Such  drafts  are  known  as  hank  money  orders^ 
and  are  usually  charged  for  at  the  same  rate  as  post  office 
or  express  money  orders. 

119.  Exchange  by  Wire.  Telegraph  companies,  for  a  fixed 
fee  in  addition  to  the  cost  of  sending  the  necessary  tele- 
grams, instruct  payees  by  wire  to  call  at  a  designated  tele- 
graph office  to  receive  a  certain  sum  of  money,  and,  in  the 
same  way,  direct  the  manager  of  such  office  to  pay  the  amount 
to  a  particular  person  on  demand. 

Express  companies  also  order  money  paid  by  telegram,  in- 
structing their  agent  at  a  certain  place  to  deliver  to  the 
payee  a  given  amount. 

Telephone  companies  direct  their  managers  at  other  places 
to  pay  given  amounts.  Banks,  too,  will  order  the  payment 
of  money  by  telegraph. 

Exchange  by  wire  is  necessarily  more  expensive,  as  the 
fees  are  quite  large.  Such  a  method  is  intended  for  use 
only  in  emergencies. 

120.  Foreign  Exchange. 

Foreign  Coins 

English  Money.     There  are :       4  farthings  in  1  penny  (<f.) 

12  pence  in  1  shilling  (s.) 
20  shilling  in  1  pound  (X) 

French  Money.      There  are :     10  centimes  in  1  decime 

10  decimes  in  1  franc 

German  Money.    There  are :  100  pfennigs  in  1  mark 

Spanish  Money.     There  are :  100  centesimos  in  1  peseta 

Note. — France,  Belgium,  Greece,  Italy,  and  Switzerland  constitute 
what  is  known  as  the  "  Latin  Monetary  Union,"  and  their  coins  are 


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MEASURES  OF  VALUE  145 

alike  in  weight  and  fineness,  occasionally  differing,  however,  in  name. 
Most  of  the  Central  and  South  American  states  possess  a  standard  coin. 

121.  Bills  of  Foreign  Exchange.  Drafts  payable  in  foreign 
countries  are  either  banker's  or  commercial  bills  of  exchange. 
They  are  based  upon  credits  in  the  foreign  cities  in  the  same 
way  as  are  bank  drafts  upon  cities  in  our  own  country,  the 
latter  being  known,  by  way  of  distinction,  as  domestic  ex- 
change. Banker's  bills  were  formerly  drawn  in  duplicate  or 
triplicate,  each  copy  being  sent  by  a  different  route  for  safety. 
Rapid  and  safe  mail  service  renders  this  no  longer  necessary, 
particularly  between  Europe  and  America.  Accordingly, 
foreign  bills  are  now  usually  drawn  singly,  the  same  as 
domestic  drafts. 

All  foreign  bills  are  payable  in  the  money  of  the  country 
upon  which  they  are  drawn,  and  quotations  of  prices  of 
foreign  exchange  give  the  cost  in  our  money  of  one  mone- 
tary unit  of  such  country.  Thus,  an  English  pound  sterling 
may  be  quoted  as  $4.86  or  14.88,  etc. 

The  ebb  and  flow  of  commerce,  with  the  balance  of  trade 
shifting  from  one  country  to  another,  causes  changes  in  the 
value  of  credits  in  one  country  which  may  be  owned  in 
another,  and  these  fluctuations  are  shown  in  daily  quotations 
issued  by  banks  and  clearing  houses.  When  there  is  a  larger 
foreign  credit  than  is  needed  for  the  demands  of  business, 
then  foreign  exchange  is  at  a  discount ;  when  less  than  busi- 
ness requires,  it  is  at  a  premium. 

Certain  banks  in  larger  cities  maintain  reciprocal  relations 
as  correspondents  with  banks  in  certain  foreign  centers. 
Banks  in  smaller  places  often  arrange  to  draw  on  such  foreign 
correspondents  through  these  larger  banks,  paying  therefor 
as  per  quotations  furnished. 

122.  Letters  of  Credit.  Ba-nks  also  issue  letters  of  credit 
to  persons  who  wish  to  travel  abroad.     They  are  addressed 


146  ELEMENTS  OF  BUSINESS  ARITHMETIC 

to  any  of  a  list  of  correspondent  banks  in  various  cities  in 
Europe,  requesting  them  to  furnish  funds  to  the  order  of  the 
holder  up  to  a  certain  limit.  Upon  reaching  one  of  the  men- 
tioned cities,  a  draft  drawn  by  the  holder  of  the  letter  for  an 
amount  within  the  limit  will  be  cashed  by  the  correspondent 
bank.  The  amounts  drawn  are  indorsed  on  the  letter  by  the 
banks  cashing  the  drafts,  and  the  letter  itself  accompanies 
the  last  draft  which  exhausts  the  limit  of  the  credit.  The 
advantage  of  these  letters  is  in  one  being  able  to  draw  varying 
amounts,  according  to  his  needs,  in  a  large  number  of  places. 

123.  Foreign  Money  Orders.  Foreign  money  orders^  pay- 
able in  the  currency  of  the  principal  countries  of  the  world, 
may  be  purchased  of  the  United  States  government  at  any 
money-order  office. 

Some  express  companies  issue  money  orders  or  drafts  on 
important  cities  in  foreign  countries,  and  these  drafts  are 
payable  in  the  money  of  such  country  at  the  company's 
agencies,  or  at  certain  banks  with  which  they  have  arrange- 
ments as  correspondents. 

Travelers''  checks^  issued  in  certain  denominations,  with 
the  foreign-money  values  in  various  countries  printed  on 
the  face  thereof,  may  be  purchased  in  any  number.  These 
checks  are  payable  at  various  banks  and  agencies,  and  pass 
current  in  almost  any  country  for  their  face  value  in  pay- 
ment of  ordinary  expenses.  A  supply  of  these  checks,  in 
different  denominations,  forms  a  very  convenient  kind  of  ex- 
change for  travelers. 

Some  express  companies  also  issue  letters  of  credit^  en- 
titling the  holder  to  draw  money  or  travelers'  checks  thereon 
up  to  a  certain  limit.  For  such  a  letter  of  credit  either  cash 
must  be  deposited,  or  it  must  be  secured  either  by  a  deposit 
of  securities  or  the  guarantee  of  a  responsible  bank,  trust 
company,  or  banker. 


MEASURES  OF  VALUE  147 

Oahle  Transfers.  Money  may  be  paid  by  cable  in  the  same 
way  as  by  telegraph. 

Shipments  of  gold  coin  or  bullion  are  frequently  made  to 
restore  a  balance  of  credits  in  foreign  exchange. 

Miscellaneous  Measures 

124.  Tradesman's  Table. 

There  are :       12  units  in  1  dozen  (doz.) 

12  dozen  in  1  gross  (gr.) 

12  gross  in  1  great  gross 
Twenty  units  are  often  called  a  score. 

125.  Stationer's  Table. 

Paper,  particularly  of  the  finer  kinds,  is  commonly  put  up 
and  sold  according  to  the  following  units : 

There  are :  24  sheets  in  1  quire 

20  quires  in  1  ream  (480  sheets) 
2  reams  in  1  bundle 
5  bundles  in  1  bale 

Paper  houses  are  gradually  adopting  the  commercial  ream 
of  500  sheets,  particularly  for  book  and  print  papers.  The 
larger  units  are  disappearing  from  use. 

PROBLEMS 

1.  The  United  States  paid  Spain  $20,000,000  for  the  Philippines. 
What  was  the  face  of  the  bill,  if  the  rate  of  exchange  was  19.38  cents? 

2.  A  merchant  owes  a  bill  of  £300  85.  10c?.  in  London.  What  will  it 
cost  him  for  a  bill,  if  the  rate  of  exchange  is  $4,865? 

3.  A  Boston  importer  owed  a  Dresden  manufacturer  26,450  marks,  and 
paid  him  by  a  bill  of  exchange.  If  the  rate  was  23.85  cents,  what  did  it 
cost  him  ? 

4.  I  owe  4250  francs  in  Paris.  If  the  rate  is  19.38  cents,  what  wiU  it 
cost  me  to  pay  the  debt  ? 


148  ELEMENTS  OF  BUSINESS  ARITHMETIC 

5.  A  merchant  bought  4  cases  musical  instruments  in  Berlin,  amount- 
ing to  3598.6  marks,  and  received  a  discount  of  i^.  If  the  exchange  is 
worth  23.87  cents,  what  will  the  bill  of  exchange  cost? 

6.  What  will  be  the  value  of  a  sight  draft  on  Berlin,  amounting  to 
3598.6  marks,  if  the  exchange  is  worth  23.88  cents  ? 

7.  What  must  be  paid  for  a  bill  on  Rome  for  6750  lire,  at  19  ^  ? 


XII 

FRENCH  METRICAL  SYSTEM 

127.  The  Metric  System.  The  rapidly  expanding  use  of 
the  measures  of  the  P'rench  Metrical  System  renders  a 
knowledge  of  its  units  more  and  more  a  necessity  in  a  busi- 
ness education.  Its  use  is  required  in  some  of  the  depart- 
ments of  the  government  and  is  authorized  in  others. 
Government  comparisons  of  standards  are  made  by  reference 
to  the  international  metric  units.  The  metric  system  is  the 
legal  standard  in  the  Philippines  and  Porto  Rico.  So  ex- 
tensive is  its  use  in  foreign  countries  that  manufacturers, 
particularly  of  scientific  instruments  and  machinery,  are  put- 
ting out  their  products  in  metric  sizes  for  the  export  trade. 

The  metric  system  is  founded  primarily  upon  the  meter^ 
which  is  the  unit  of  length.  All  other  units  are  so  related 
to  this  as  greatly  to  simplify  calculations  in  which  weights 
and  measures  are  involved.  It  is  this  simple  relation  to  one 
fundamental  unit  which  makes  it  a  system.  Add  to  this  the 
fact  that  it  is  a  decimal  system,  and  its  extreme  simplicity 
both  in  plan  and  ease  of  calculation  is  explained. 

With  a  desire  to  found  the  unit  upon  something  fixed,  the  French 
mathematicians  took  one  ten-thousandth  part  of  the  earth's  quadrant  for 
a  Kilometer,  of  which  the  more  usable  meter  is  a  thousandth  part.  Under 
the  direction  of  an  International  Bureau  of  Weights  and  Measures,  es- 
tablished near  Paris  by  the  principal  nations  of  the  world,  standard 
meters  and  kilograms  were  cast  from  platinum-iridium,  and  distributed 
to  the  various  nations.  By  the  law  of  1893,  all  our  common  units  are 
derived  from  these  international  standards,  which  are  kept  in  the  office 
of  the  National  Bureau  of  Standards  at  Washington. 

149 


150  ELEMENTS  OF  BUSINESS  ARITHMETIC 

The  plan  of  naming  the  derived  units  is  equally  simple. 
Derivatives  from  the  Greek,  viz.  deka  for  ten,  hehta  for 
hundred,  hilo  for  thousand,  and  myria  for  million,  are  pre- 
fixed to  the  name  of  the  units  for  all  multiples,  while  the 
fractional  units  are  indicated  by  the  Latin  prefixes,  deei  for 
tenth,  centi  for  hundredth,  and  milli  for  thousandth. 

A  practical  knowledge  of  the  metric  system  should  be 
acquired  by  a  study  and  use  of  the  measures  themselves. 
Each  unit  studied  should  be  fixed  by  actual  use,  without 
regard  to  the  nearest  English  unit,  and  the  power  to  esti- 
mate accurately  in  its  terms  developed  through  drill.  After 
such  knowledge  of  the  various  units  has  been  acquired,  and 
one  is  able  to  think  in  these  units,  a  study  of  comparative 
values  between  them  and  the  English  units  has  its  value. 
Comparative  tables  are  appended  for  reference  only. 

MEASURES  OF  LENGTH 

128.  The  Unit  of  Length.  As  before  stated,  the  meter  is 
one  thousandth  part  of  a  kilometer,  which  is  one  ten-thou- 
sandth part  of  the  earth's  quadrant.  It  is,  therefore,  one  ten- 
millionth  of  the  distance  from  the  equator  to  the  pole.  From 
it  all  the  other  units  of  the  French  Metrical  System  are 
derived.  The  surveyor's  chain  is  a  dekameter  or  half  deka- 
meter  in  length. 

Table 
There  are : 

10  millimeters  (mm.)  in  1  centimeter   (cm.) 
iO  centimeters  in  1  decimeter     (dm.) 

10  decimeters  in  1  meter  (i^O 

10  meters  in  1  dekameter    (Dm.) 

10  dekameters  in  1  hektameter  (Hm.) 

10  hektameters  in  1  kilometer     (Km.) 


FRENCH  METRICAL  SYSTEM  151 

MEASURES  OF  AREA 

129.  Unit  of  Area.  The  unit  of  area  is  the  square  meter. 
In  a  like  manner  each  denomination  of  linear  measure  is 
squared,  forming  the  units  of  area.  A  square  meter  being 
10  decimeters  wide  and  10  decimeters  long,  has  100  square 
decimeters.  The  same  principle  is  applied  to  the  other 
units. 

There  are:  ^^^^^ 
100  sq.  millimeters  (sq.  mm.)  in  1  sq.  centimeter   (sq.  cm.) 

100  sq.  centimeters  in  1  sq.  decimeter     (sq.  dm.) 

100  sq.  decimeters  in  1  sq.  meter             (sq.  m.) 

100  sq.  meters  in  1  sq.  dekameter    (sq.  Dm.) 

100  sq.  dekameters  in  1  sq.  hektameter  (sq.  Hm.) 

100  sq.  hektameters  in  1  sq.  kilometer     (sq.  Km.) 

The  chief  use  of  the  larger  area  units  is  in  the  measure- 
ment of  land  surfaces.  For  that  purpose,  the  square  deka- 
meter is  given  a  special  name,  and  is  known  as  the  are.,  and 
the  square  hektameter  as  the  hectare.  The  latter  is  the 
more  commonly  used  land -area  measure,  while  the  square 
kilometer  has  no  practical  application. 

MEASURES  OF  VOLUME 

130.  Units  of  Volume.  The  units  of  volume  are  derived 
in  the  same  way,  each  linear  unit  being  cubed.  A  cubic 
meter  being  10  decimeters  along  each  edge,  contains  1000 
cubic  decimeters.  Likewise,  there  are  1000  of  each  cubic 
denomination  in  the  next  higher  one. 

There  are:  ^^^^^ 

1000  cu.  millimeters  (cu.  mm.)  in  1  cu.  centimeter  (cu.  cm.) 
1000  cu.  centimeters  in  1  cu.  decimeter  (cu.  dm.) 

1000  cu.  decimeters  in  1  cu.  meter  (cu.  m.) 

1000  cu.  meters  in  1  cu.  dekameter  (cu.  Dm.) 


152  ELEMENTS  OF  BUSINESS  ARITHMETIC 

131.  Capacity  Units.  For  measuring  capacity,  the  cubic 
decimeter  is  used.  The  special  name  of  liter  is  given  to  it, 
from  which  are  derived  the  denominations  of  capacity  with 
the  constant  ratio  of  ten  between  successive  units,  and  desig- 
nated by  the  usual  prefixes. 

There  are :  Table 

10  milliliters  (ml.)  in  1  centiliter    (cl.) 
10  centiliters  in  1  deciliter    (dl.) 

10  deciliters  in  1  liter  (1.) 

10  liters  in  1  dekaliter    (Dl.) 

10  dekaliters  in  1  hektaliter  (HI.) 

Wood  Measure.  For  the  measure  of  wood,  the  cubic  meter 
is  given  a  special  name,  the  stere.  This,  likewise,  may  be 
treated  decimally  and  a  series  of  denominations  formed. 

MEASURES  OF  WEIGHT 

132.  Unit  of  Weight.  For  a  unit  of  weight,  one  cubic 
centimeter  of  distilled  water  is  taken,  at  its  greatest  density, 
in  the  latitude  of  Paris  and  at  sea  level.  It  is  called  a  gram. 
Decimal  denominations  are  derived  from  this  in  the  same 
way  as  from  the  liter.  A  cubic  decimeter  or  liter  of  pure 
water  contains  1000  cubic  centimeters,  and  hence  weighs 
1000  grams,  or  one  kilogram.  For  very  heavy  articles  the 
weight  of  a  cubic  meter  of  water  is  used,  or  1000  kilograms, 
and  this  is  called  a  tonneau. 

There  are :  Table 

10  milligrams  (mg.)  in  1  centigram  (eg.) 

10  centigrams  in  1  decigram  (dg.) 

10  decigrams  in  1  gram  (g.) 

10  grams  in  1  dekagram  (Dg.) 

10  dekagrams  in  1  hektagram  (Hg.) 

10  hektagrams  in  1  kilogram  (Kg.) 

100  kilograms  in  1  tonneau  (T.) 


FRENCH  METRICAL  SYSTEM  153 

133.  Decimal  Methods.  As  the  successive  units  of  the 
metric  system  usually  bear  the  ratio  of  ten  to  each  other, 
various  denominations  are  very  conveniently  written  as  a 
decimal  of  one  denomination.     Thus,  3  Dm.  5  m.  6  dm.  7  cm. 

5  mm.  would  usually  be  written  as  35.675  m. 

For  the  same  reason  a  change  from  one  denomination  to 
another  may  be  accomplished  by  merely  moving  the  decimal 
point.  Thus,  the  above  35.675  m.  may  also  be  written  as 
3.5675  Dm.,  or  356.75  dm.,  etc.  Obviously  this  ease  of 
reduction  is  a  decided  advantage  in  the  use  of  the  metric 
system. 

PROBLEMS 

1.  Write  5463  cm.  as  kilometers;    as  decimeters;    as  dekameters; 
as  meters. 

2.  Write  5360  sq.  m.  as  ares ;  as  hektares. 

3.  Write  5200  cl.  as  liters;  as  hektaliters. 

4.  At  6  ^  a  meter,  what  will  it  cost  to  build  a  fence  54  Dm.  6  m. 
long? 

5.   How  many  rails  9  m.  4  dm.  long  will  it  take  to  build  a  railway 
20  Km.  4  Hm.  2  Dm.  long? 

6.  What  is  the  weight  of  a  liter  of  water  in  grams  ?    What  is  the 
weight  of  a  cubic  meter  of  water  ?    How  many  liters  in  it  ? 

7.  A  circular  lot  is  27  meters  in  diameter.     What  is  the  area  in 
ares? 

8.  How  many  ares  in  a  rectangular  field  8  Dm.  long  and  5  Dm.  4  m. 

6  dm.  wide? 

9.  How  many  cubic  meters  of  water  in  a  tank  10  m.  long,  6  m.  7  dm. 
high,  and  8  m.  wide  ? 

10.  A  fence  is  5  boards  high  and  12  Hm.  2  Dm.  long.  How  many 
boards  3  m.  long  are  there  in  it  ? 

11.  How  much  carpet  1  m.  wide  will  be  needed  to  cover  a  room  6.4  m. 
long  and  5.5  m.  wide  ? 

12.  What  will  50  1.  of  mercury  weigh,  if  it  is  13.5  times  heavier  than 
water? 


154  ELEMENTS  OF  BUSINESS  ARITHMETIC 

13.  If  a  pile  of  wood  is  32  m.  long,  5  m.  2  dm.  wide,  and  3  m.  6  dm. 
high,  what  is  it  worth  at  $2  a  stere? 

14.  How  much  wheat  will  a  bin  4  m.  long,  3  m.  4  dm.  wide,  and  2  m. 
high  hold?    What  is  the  value  at  60;*  a  liter?     What  is  the  weight ? 

15.  How  many  tiles  40  cm.  x  20  cm.  will  be  used  in  tiling  a  floor  9  m. 
6  dm.  long  and  5  m.  4  dm.  wide? 

16.  If  copper  is  8.8  times  as  heavy  as  water,  what  is  the  weight  of 
8  cu.  dm.  of  the  metal? 

17.  What  will  it  cost  to  plaster  the  walls  and  ceiling  of  a  room  that 
is  6  m.  5  dm.  by  5  m.  8  dm.  by  4  m.  2  dm.,  at  32;*  a  square  meter? 

18.  What  will  it  cost  to  carpet  a  room  5  m.  4  dm.  by  3  m.  2  dm.  with 
carpet  8  dm.  wide  at  80;*  a  meter? 

19.  What  will  it  cost  to  paint  the  walls  of  a  barn  15.5  m.  long,  10  m. 
wide,  and  9.5  m.  high,  at  35;*  per  square  meter? 

20.  Find  the  capacity  of  a  tank  5  m.  long,  3  m.  6  dm.  wide,  and  2  m. 
6  dm.  high. 

21.  What  will  be  the  cost  of  4608  Kg.  of  hay  at  $  12  a  ton  ? 

22.  How  many  steres  of  wood  in  a  pile  10  Dk.  long,  3  m.  4  dm.  wide, 
and  1.5  m.  high? 

23.  Gold  is  19.5  times  as  heavy  as  water.  What  is  the  weight  of  a 
cubic  centimeter  of  gold  ? 

24.  How  many  jars,  each  containing  2.5  liters,  can  be  filled  from  a 
cask  containing  145.5  DL? 

25.  A  room  6.3  m.  long  and  4  m.  wide  will  require  how  many  meters 
of  carpet  8  dm.  wide  to  cover  it? 

26.  Find  the  area  of  the  four  walls  of  a  room,  10.5  m.  long,  6.5  m. 
wide,  and  5.4  m.  high.  Area  of  ceiling?  Cost  to  plaster  the  room  at 
35  ;*  a  square  meter  ? 

27.  What  will  it  cost  to  paint  the  walls  and  ceiling  of  a  room  8.5  m. 
X  5.4  m.  X  4  m.,  deducting  for  four  windows  each  2  m.  x  1  m.,  at  15^  per 

square  meter  ? 

28.  A  room  is  8  m.  x  5  m.  x  3.2  m.  Deducting  for  five  windows, 
each  2.1  m.  x  1  m.,  2  doors  each  2.8  m.  x  1.4  m.,  and  a  baseboard  2  dm. 
high,  what  will  be  the  cost  of  plastering,  at  45^  a  square  meter?  The 
cost  of  paper  4  dm.  wide  at  $4.20  per  roll  of  10  meters?  What  is  the 
cost  of  carpet  5.2  dm.  wide,  at  $1.50  per  meter? 


FRENCH  METRICAL  SYSTEM 


155 


134. 


lin. 
1ft. 
1yd. 
1  mi. 


1  cu.  in. 
1  cu.  ft. 
1  cu.  yd. 
1  bushel 


COMPARATIVE  TABLES 

1.    Customary  Units  to  Metric  Units. 


Length 

25.4001  mm. 
.304801  m. 
.914402  m. 
1.60935  Km. 

Volume 

=  16.387  cu.  cm. 
=  .02832  cu.  m. 
=  .765  cu.  m. 
=  .35239  HL 


Area 

1  sq.  in.   =  6.452  sq.  cm. 
Isq.  ft.    =9.290  sq.  dm. 
1  sq.  yd.  =  .836  sq.  m. 
1  acre       =  .4047  Hm. 


1  fl.  dr. 
1  fl.  oz. 
1  qt. 
IgaL 


Capacity 
=  3.70  cu.  cm. 
=  29.57  mm. 
=  .94636  1. 
=  3.78543  L 


Weight 

1  gr.  =  64.7989  mg. 

1  av.  oz.    =  28.3495  g. 
1  av.  IK    =  45359  Kg*  ^ 
1  troy  oz.  =  31.10348  g. 

2.    Metric  Units  to  Customary  Units. 

Length  Square 

1  sq.  m.  =  1550  sq.  in. 
sq.  m.  =  10.764  sq.  ft. 
sq.  m.  =  1.196  sq.  yd. 


1  m.      =  39.3700  in. 
1  m.     =  3.28083  ft. 
1  m.     =  1.093611  yd 
1  Km.  =  o62137  mi. 


1 
1 
IHa. 


=  2.471  A. 


1  cu.  cm. 
1  cu.  dm. 
1  cu.  m. 
1  cu.  m. 


Cubic 

=  .0610  cu.  in. 
=  61.023  cu.  in. 
=  35.314  cu.  ft. 
=  1.308  cu.  yd. 


Capacity 
1  mm.  =  .27  fl.  dr. 
1  cl.      =  .338  fl.  oz. 
11.       =  1.0557  qt. 
1  Dl.    =  2.6417  gaL 
IHL    =  2.8337  bu. 


156  ELEMENTS  OF  BUSINESS  ARITHMETIC 


Weight 


1  mg.  =  .01543  gr. 
1  Kg.  =  15432.36  gr. 
1  Hg.  =  3.5274  av.  oz. 
1  Kg.  =  2.20462  av.  lb. 
1  T.     =  2204.6  av.  lb. 


XTII 
PERCENTAGE 

135.  Meaning  and  Use.  By  per  cent  is  meant  hundredths. 
The  term  is  an  abbreviation  of  the  Latin  per  centum^  or  "  by 
the  hundred."     As  a  further  abbreviation,  %  is  used. 

Just  as  decimals  are  a  development  of  certain  forms  of 
fractions  having  advantages  in  ease  of  operation,  so  percent- 
age is  the  development  of  a  certain  kind  of  decimal  fraction, 
having  advantages  for  comparison  in  business  transactions. 
A  wide  range  of  variation  in  the  size  of  different  parts  of  a 
whole  may  be  indicated  by  hundredths,  while  operations  are 
often  made  easy  by  frequent  opportunities  for  using  simple 
fractions.  Since  hundredths  may  be  written  as  decimals, 
operations  in  percentage  also  possess  all  the  advantages  of 
decimals  in  ease  and  quickness  of  computation. 

In  Sees.  19  and  20  the  methods  of  finding  any  number  of 
hundredths  were  developed.  In  the  treatment  of  simplified 
processes  under  Fractional  Parts  (Chapter  IV),  suggestions 
were  made  for  shortening  those  methods.  While  not  enter- 
ing now  upon  the  study  of  anything  essentially  new,  a 
further  and  more  systematic  study  of  hundredths  must  be 
made  that  we  may  better  understand  percentage  in  its  vari- 
ous applications  to  business. 

136.  Profit  and  Loss.  Business  enterprises  are  carried  on 
in  the  expectation  of  making  a  profit  for  those  who  invest 
money  or  other  wealth  in  them.  At  regular  intervals,  usu- 
ally every  year,  a  careful  inventory  is  made  of  the  results  of 
the  year's  business,  to  find  out  how  much  profit  has  been  made. 

167 


158  ELEMENTS  OF  BUSINESS  ARITHMETIC 

It  sometimes  happens  that  such  an  inventory  shows  that  the 
business  has  been  carried  on  at  a  loss.  Whether  a  loss  or  a 
profit,  the  amount  is  first  ascertained,  and  then,  for  purposes 
of  comparison,  the  per  cent  that  the  loss  or  profit  is  of  the 
capital  invested  is  found. 

Profit  or  loss  is  expressed,  then,  in  terms  of  per  cent. 
Because  of  the  simplicity  of  the  ideas  involved,  problems  in 
profit  and  loss  are  used  in  the  study  of  percentage  processes. 

137.  Finding  50%,  25  %,  or  20  %.  Since  there  are  one  hun- 
dred hundredths  in  the  whole  of  anything,  100  %  of  anything 
is  equal  to  the  whole  of  it.  If  100  %  of  anything  is  all  of  it, 
50  %  is  ^  of  it.  Likewise,  since  J  of  100  %  equals  25  %,  then 
25  %  of  anything  is  ^  of  it ;  20  %  of  anything  is  ^  of  it,  etc. 

What  per  cent  of  anything  is  |  of  it  ?  What  f  ?  What 
I?     Whatf? 

Summary 

100%  =1^8-  25%  =1  75%  =1  60%  =1 

50%=i  20%=i  40%  =  |  80%  =  | 

ILLUSTRATIVE  PROBLEMS 

1.  Mr.  King  invested  $  400  in  grain,  and  lost  50  %.  How  much  did 
he  lose  ? 

Since  50%  of  anything  is  ^  of  it,  then  his  loss  is  i  of  $400,  or 
$200. 

Stated  thus : 

50%=  h 
i  of  $400  =  $200 

2.  A  clerk  spends  $  30,  which  is  25  %  of  his  salary.  What  is  his 
salary  ? 

Since  25  %  of  anything  is  i,  then  $  30  is  J  of  his  salary,  and  f  are  4 
times  $  30,  or  $  120,  his  salary. 

Statement  : 

25%=  i 
J  of  salary  =  $  30 

$  30  X  4  =  $  120,  salary. 


PERCENTAGE  159 

3.  A  man  having  $  75,  spends  $  15  for  a  coat.  What  per  cent  of  his 
money  does  he  spend  ? 

Since  he  has  $  75,  and  spends  $  15,  he  spends  f|,  or  |  of  his  money. 
But  ^  of  anything  is  20  %  of  it ;  therefore  he  spends  20  %  of  his  money 
for  the  coat. 

Statement  : 

H  =  i 

^  =  20  %  spent  for  coat. 

4.  A  man  owns  300  A.  of  land,  which  is  50%  more  than  he  owned 
the  year  before.     How  much  did  he  own  the  year  before  ? 

Since  50  %  of  anything  is  |  of  it,  he  would  have  ^  more,  or  1^  times 
what  he  had  the  year  before.     But  1^  times  anything  is  |  of  it ;  there- 
fore 300  A.  is  f  of  what  he  had,  and  ^  is  ^  of  300  A.,  or  100  A.     The 
whole  of  what  he  had,  then,  was  twice  100  A.,  or  200  A. 
Statement  : 

50%  =  ^. 

I  of  what  he  had  =  300  A. 

^  =  ^  of  300  A.,  or  100  A. 

f  =  100  A.  X  2,  or  200  A.,  what  he  had  the  year  before. 

5.  I  sell  a  house  for  $  600  and  lose  20  %  by  doing  so.  What  did  the 
house  cost  ? 

Since  20%  of  anything  is^  of  it,  I  have  lost  ^,  and  have  f  remaining. 
Then,  $  600  is  f  of  the  cost,  and  ^  is  i  of  $  600,  or  $  150.  The  whole  cost, 
then,  would  be  5  times  $150,  or  $750. 

Statement  : 

20%  =  i 
4  of  cost  =  $600 

i  =  iof  $600,  or$150 

I  =  $  150  X  5,  or  $  750,  cost. 

PROBLEMS 

^1.    A  farmer  invested  $600  in  hogs  and  lost  50  7o-     How  much  did 

he  lose  ? 

V    2.   A  bookkeeper  spends  $28,  which  is  25%  of  his  monthly  salary. 

What  is  his  salary  ? 

^  3.   A  man  owns  600  sheep,  which  is  50%  more  than  he  owned  the 

year  before.     How  many  did  he  own  the  year  before  ? 

^   4.   A  horseman  bought  a  horse  for  $160  and  sold  it  for  $200.     What 

per  cent  did  he  gain  ? 


160  ELEMENTS  OF  BUSINESS  ARITHMETIC 

5.  A  farmer  had  150  hogs  and  lost  20  %  of  them.     How  many  did 
he  have  left  ? 

6.  An  agent  buys  books  for  $3.25  and  sells  them  for  $  6.50.    What 
per  cent  did  he  gain  ? 

7.  A  firm  buys  lots  for  $520  and  sells  them  for  $650.    What  per 
cent  is  the  gain  ? 

8.  A   man  paid  $  80  for  a  horse  and  sold  it  for  $  120.     What  per 
cent  did  he  gain  ? 

*  9.  A  poultry  man  sold  240  chickens,  which  were  |  of  his  flock. 
How  many  were  there  in  the  flock?  What  per  cent  of  the  flock  did  he 
sell? 

it  10.  Mr.  Frank  paid  $  80  for  a  horse.  What  per  cent  of  his  money 
did  he  pay  if  he  had  $200? 

11.  I  sell  a  farm  for  $  6000  and  gain  25  %  by  doing  so.     What  did  the 
farm  cost  me  ? 

12.  I  sell  a  horse  for  $  600  and  lose  20  %  by  doing  so.     What  did  the 
horse  cost  me  ? 

^13.  A  farmer  lost  60  hogs  by  disease,  which  was  40%  of  his  herd. 
How  many  had  he  at  first? 

14.  A  land  agent  had  1600  acres  of  land  and  sold  60%  of  it.     How 
many  acres  did  he  sell  ? 

15.  A  capitalist  gave  $  5400,  which  was  75  %  of  the  amount  he  gained, 
to  a  public  library.     What  did  he  gain? 

16.  A  broker  sold  30  shares  of  stock,  which  was  20  %  of  all   he  had. 
How  many  shares  did  he  have  ? 

17.  If  A's  money,  $42,  is  20%  more  than  my   money,   how  much 
have  I? 

18.  If  84  sheep  is  20  %  less  than  the  number  of  sheep  I  have,  how 
many  have  I? 

19.  A  farmer  buys  goods  for  $  60  and  sells  them  for  $48.     What  per 
cent  does  he  lose  ? 

138.    Finding  33^%,  16|%,  12|%,  and  14f  %. 

l-of  100%  =  33i% 
^ofl00%=16f% 
I  of  100%  =121% 
|ofl00%  =  14f% 


PERCENTAGE  161 

Then, 

331%  =  i  14|%  =  i  371%  =  3 

16|%=|  66|%  =  J  621%  =  ! 

28^%  =f 

It  is  well  to  remember  that  |  =  -J,  or  33|^  %  ;  |  =  ^,  or  50  % ; 
f  =  f,  or  66|%;  |  =  i,  or  25%;  f  =  i  or  50%;  and  f  =  f, 
or  75%. 

PROBLEMS 

1.  If  $15  is  33^%  of  my  money,  how  much  have  I? 

2.  A  has  172.    B  has  33^%  less.     How  much  has  B? 

3.  If  A  has  $48,  which  is  33^%  less  than  B*s  money,  how  much 
hasB? 

4.  Mr.  Hartman  buys  a  buggy  for  $63,  which  is  12|%  less  than  he 
paid  for  a  horse.     What  did  the  horse  cost? 

5.  C  has  70  acres  of  land,  which  is  16f  %  as  much  as  B's.    How  much 
hasB? 

6.  A  drove  21  miles  in  a  day,  which  was  16|%  farther  than  B  drove. 
How  far  did  B  drive  ? 

7.  A  merchant  buys  goods  for  $480  and  sells  them  for  $540.     What 
is  his  per  cent  of  gain  ? 

t^e.   A  broker  buys  stock  for  $4800  and  sells  it  for  $5600.     What  per 
cent  did  he  gain  ? 

9.   Mr.  Jacobs  buys  goods  for  $39  and  sells  them  for  $52.     What 
per  cent  did  he  gain  ? 

10.   Mr.  Gould  buys  a  lot  for  $1200  and  sells  it  for  $2400.    What  per 
cent  did  he  gain  ? 

>^11.   A  raised  80  bushels  of    potatoes,  which  was  16|%  less  than  B 
raised.     How  many  bushels  did  B  raise? 

12.  If  I  buy  a  horse  for  $120  and  sell  it  for  $140,  what  per  cent  do 
I  make  ? 

13.  A  received  a  salary  of  $720  and  spends  66|%  of  it.     How  much 
did  he  spend? 

14.  If  $49  is  874%  of  my  wages,  what  are  my  wages?         y^^ 


162  ELEMENTS  OF  BUSINESS  ARITHMETIC 

15.  If  37^%  of  my  money  is  |33,  how  much  have  I? 

16.  A  merchant  makes  a  profit  of  $4200  from  his  business  and  spends 
33 1  %  of  it  for  family  expenses.     How  much  does  he  save? 

17.  A  merchant  sold  24  dozen  eggs,  which  was  37^%  of  all  he  had. 
How  many  had  he  ? 

18.  A  farmer  had  24  hogs  in  one  pen  and  12  in  another.  He  took  4 
hogs  from  the  first  and  put  them  in  the  second.  What  per  cent  decrease 
in  the  first  pen  ?     What  per  cent  increase  in  the  second  ? 

19.  He  again  took  from  the  first  4  hogs  and  put  them  in  the  second. 
What  per  cent  increase  in  the  second  and  what  decrease  in  the  first? 

20.  Books  that  cost  $21  were  sold  so  as  to  gain  28f%.  Find  the 
selling  price? 

f^  21.   Goods  that  cost  $88  were  sold  so  as  to  gain  $77.     What  is  the 
per  cent  of  gain? 

139.    Finding  10  %,  1  %,  5  %,  and  \  %. 

Then, 


Jj    of  100%  =  10% 

10%=tV 

30%=T% 

^^oflOO%=l% 

1%    =Tb 

70%=Jj 

2V    of  100%  =  5% 

5%    =2V 

90%  =  ^^,  etc, 

^ofl00%=i% 

\1o    =jk 

By  Sec.  19,  we  learned  that  to  find  J^,  we  should  point 
off  one  decimal  place  or  remove  the  decimal  point  one  place 
to  the  left ;  to  find  ^-J^j,  we  remove  the  decimal  point  two 
places  to  the  left.  Then  to  find  10  %  or  1  %  of  any  number, 
we  remove  its  decimal  point  respectively  one  or  two  places 
to  the  left.  Thus,  10%  of  33  is  3.3,  and  10%  of  §54.20 
is  15.42.  Also,  1%  of  256  is  2.56,  and  1%  of  15425  is 
154.25. 

Since  ^^  is  \  of  J^,  we  may  find  ^,  or  5%,  by  taking  \  of 
Jjjofit.  Thus,  5%  of  $976  is  1  of  §97.60,  or  $48.80.  Like- 
wise, -|-%  or  glo  is  1  of  1%,  or  \  of  -j^^.  Thus,  \%  of 
1976.25  is  1  of  §9.7625,  or  §4.88. 


PERCENTAGE  163 

PROBLEMS 

1.  A  miller  makes  56  bushels  of  wheat  into  flour  in  one  day.  If 
this  is  10  %  of  the  wheat  he  has,  how  many  days  will  it  take  him  to  grind 
all  of  it  at  the  same  rate  ? 

2.  James  has  $76,  which  is  5%  less  than  I  have.    How  much  have  I? 

3.  A  man  owed  $6700  when  he  died.  His  property  was  worth 
90  %  of  this  amount.     What  was  the  value  of  his  property  ? 

4.  Mr.  Berry  sold  a  carriage  for  $195,  which  was  30%  more  than  it 
cost  him.     What  did  it  cost  him  ? 

5.  A  merchant  sold  goods  that  cost  him  $34  at  a  loss  of  1%.  Find 
the  selling  price. 

6.  A  house  worth  $6000  is  rented  for  5%  of  its  valuation.  What  is 
the  rent? 

7.  A  bedroom  suite  was  sold  for  $77,  which  was  10%  above  cost. 
Find  the  cost. 

8.  A  farmer  sold  two  cows  for  $  38  each,  gaining  5  %  on  one  and  losing 
5%  on  the  other.  Find  the  cost  of  each.  Find  the  loss  or  gain  in  the 
transaction. 

9.  An  implement  dealer  buys  a  wheat  drill  for  $54,  and  sells  it  at 
a  profit  of  10%.     What  is  the  selling  price  ? 

10.   A  young  man  had  $600  and  received  $4200  by  will.    What  per 
cent  was  the  increase  in  his  wealth? 
^  11.   A  ton  of  coal  was  sold  for  $9,  which  was  a  gain  of  12|%.     Find 
the  cost. 

12.  A  father  had  $1400.  He  gave  25%  to  his  son,  40%  of  the  re- 
mainder to  his  elder  daughter,  and  the  remainder  to  the  younger  daughter. 
How  much  did  each  receive  ? 

13.  A  merchant  sells  $56  worth  of  goods  to-day,  which  is  12^%  less 
than  he  sold  the  day  before.  Find  amount  of  his  sales  the  day  before. 
^14.  A  grocer  buys  a  hogshead  (56  gal.)  of  molasses  and  sells  30%  of 
it  the  first  day.     How  many  gallons  had  he  left? 

15.  Mr.  Burkett  sold  a  buggy  for  $85,  which  was  70%  more  than  it 
cost  him.     Find  cost. 

16.  I  loan  $600  and  receive  $30  interest.  What  per  cent  is  this  of 
the  amount  loaned  ? 

17.  A  commission  merchant  charges  $20  for  selling  $4000  worth  of 
wheat.     What  per  cent  is  that  of  the  selling  price  of  the  wheat  ? 


164  ELEMENTS  OF  BUSINESS  ARITHMETIC 

140.  Finding  Other  Per  Cents.  Much  the  larger  part  of 
all  business  problems  involving  percentage  has  to  do  with 
per  cents  given  in  Sections  137-139.  Most  of  the  prob- 
lems in  interest,  bank  discount,  and  trade  discount  may  be 
solved  with  them  or  with  slight  modifications  of  them,  and 
these  are  the  most  commonly  used  of  all  percentage  applica- 
tions. A  thorough  mastery  of  the  per  cents  thus  far  devel- 
oped, then,  is  of  first  importance,  and  they  should  be 
practiced  upon  until  problems  involving  any  phase  of  these 
per  cents  can  be  solved  quickly  and  with  accuracyo 

Finding  any  per  cent,  other  than  those  mentioned,  is  based 
on  the  method  of  finding  1  %.  Thus,  7,%  is  -^^  or  7  times 
1  %  ;  13  %  is  JqS.  or  13  times  1%  ;  34  %  is  -^^-^,  or  34  times 
1  %  ;  etc.  To  find  any  given  per  cent  first  find  1  %  by 
removing  the  decimal  point  two  places  to  the  left,  and  then 
multiply  by  the  number  of  per  cent  required. 

Thus,  4  %  of  13296  is  132.96  x  4  =1157.04. 

PROBLEMS 

1.  A  farmer  raised  450  bu.  of  potatoes  and  sold  6  %  of  them.  How 
many  bushels  had  he  left  ? 

2.  I  spent  $  28,  which  was  14  %  of  my  money.  How  much  had  I  at 
first? 

3.  I  have  ^65  and  spend  12 7o  of  it  for  trousers.     What  do  they  cost? 

4.  A  buys  goods  for  $36  and  sells  them  for  $39.  What  is  his  per 
cent  of  gain  ? 

5.  Mr.  Bunger  sells  his  horse  for  $65  and  by  doing  so  gains  8^%. 
What  was  the  cost? 

6.  If  Street  &  Co.  pay  $45  for  dishes  and  sell  them  for  $48,  what 
per  cent  gain  have  they? 

7.  If  a  coal  dealer  bought  coal  for  $7.50  and  sold  it  for  6|%  advance, 
what  did  he  gain  ? 

8.  Sold  goods  so  as  to  gain  $  86,  which  was  a  gain  of  12  %.  What 
was  the  cost?    What  the  selling  price? 

9.  I  sold  goods  for  $  376  and  lost  6%.     Find  the  cost. 


PERCENTAGE  165 

10.  1  have  320  sheep  and  buy  8  more.     What  per  cent  do  I  add  to  my 
flock? 

11.  I  have  $39  and  spend  1 21.     What  per  cent  do  I  spend  ? 

12.  A  merchant  paid  |17  for  goods  and  sold  them  so  as  to  gain  |3, 
What  per  cent  did  he  make  ? 

13.  A  man  spent  28%  of  his  money  for  a  coat  that  cost  $56.     How 
much  money  did  he  have  ? 

14.  A  miller  lost  36%  of  his  wheat  by  fire  and  had  1280  bushels 
remaining.     How  many  bushels  did  he  lose  ? 

15.  A  had  240  chickens  and  sold  84  of  them.     What  per  cent  of  his 
flock  had  he  left? 

16.  A  dealer  sold  grain  at  a  profit  of  16%  and  received  $696.    What 
did  it  cost  ? 

17.  A  speculator  sold  land  for  $9200,  which  was  8%  less  than  it  cost 
him.     What  did  it  cost  him  ? 

18.  A  stockman  bought  horses  for  $84  and  sold  them  for  $  96.    What 
per  cent  did  he  gain  ? 

19.  Isellgoodsfor  $184  and  lose  8%.     What  did  they  cost  ? 

20.  A  man  sold  his  horse  for  $  147,  which  was  $  33  less  than  he  paid 
for  it.     What  per  cent  did  he  lose  ? 

GENERAL  PROBLEMS 

1.  If  a  grocer  adds  a  pound  of  Java  coffee  to  every  four  pounds  of 
Mocha,  what  per  cent  of  the  mixture  is  of  each  coffee  ? 

2.  A  man  owning  |  of  a  mill,  sells  f  of  his  share.     What  per  cent  of 
the  mill  does  he  still  own  ? 

3.  A  man's  money  invested  at  10  %  annual  interest  yields  $  125  a 
month.     How  much  has  he  invested  ? 

4.  My  agent  sold  $  625  worth  of  goods  and  charged  me  \%  commis- 
sion for  selling.     How  much  money  did  I  pay  him  ? 

5.  A  mechanic  has  $42  a  month  left  after  paying  6f  %  of  his  wages 
for  car  fare.    What  are  his  wages?    How  much  does  he  pay  for  car  fare? 

6.  Parker  &  Co.  pay  $60  for  a  buggy  and  sell  it  for  $64.     What  per 
cent  do  they  make  ? 

7.  A  hardware  man  sold  a  stove  for  $84  and  lost,  by  so  doing,  6|%. 
What  was  the  cost  of  the  stove  ? 


166  ELEMENTS  OF  BUSINESS  ARITHMETIC 

8.  A  jeweler  sold  a  watch  so  as  to  gain  $2,  which  was  2J%  of  the 
cost.    What  was  the  cost  ? 

9.  The  annual  interest  on  my  money  loaned  at  4%  is  $75.     How 
much  have  I  loaned? 

10.  A  man  invested  60  %  more  money  in  a  business  than  his  partner, 
and  the  difference  between  their  investments  was  $3000.  How  much 
did  each  invest  ? 

11.  A  man  bought  a  horse  and  a  cow  for  $300,  the  horse  costing  50% 
more  than  the  cow.     How  much  did  each  cost? 

12.  I  of  James's  money  is  75  %  of  Henry's,  and  |  of  Henry's  is  25  %  of 
John's.    John  has  1 32.     How  much  has  James  ? 

13.  A  capitalist  had  a  half  interest  in  a  ranch  and  sold  |  his  interest 
for  13600.  The  sale  was  made  at  a  gain  of  25%.  What  was  the  cost  of 
the  ranch? 

14.  A  bank  building  rents  for  |4200  a  year,  which  is  12^%  of  its 
value.     What  did  it  cost,  if  it  had  increased  in  value  40  %? 

15.  If  a  firm  quits  business  with  property  worth  $2600  and  owes 
$3900,  what  per  cent  of  their  debt  can  they  pay?  How  many  cents  on 
the  dollar? 

16.  If  I  sell  $  6000  worth  of  goods  for  my  principal  and  remit  him 
$5900,  what  per  cent  commission  did  I  charge  him? 

17.  A  farmer  lost  16|%  of  his  hogs,  sold  40%  of  the  remainder,  and 
had  left  120.     How  many  had  he  at  first  ? 

18.  A  man  bought  property  which  increased  in  value  16|%  the  first 
year.  He  finally  sold  it  at  an  increase  of  2o%  over  this,  and  received 
$  8400.     What  was  the  cost  ? 

19.  A  lady  bought  a  piano  for  $480,  which  was  25%  of  the  money  she 
had  in  the  bank.  The  money  she  had  in  the  bank  was  40%  of  the 
value  of  real  estate  that  she  owned.  What  was  the  value  of  the  real 
estate  ? 

20.  A  confectioner  sold  candy  for  60  j^  a  pound.  If  this  were  25% 
more  than  it  cost  him,  what  was  the  cost? 

21.  A  clothier  sold  a  suit  of  clothes  for  $  18  and  lost  40  %.  He  then 
sold  another  at  a  profit  of  16|%  and  gJtined  as  much  as  he  had  lost  on 
the  first.     What  was  the  cost  of  each  suit? 

22.  I  sold  a  house  and  lot  for  $  1600,  losing  20  %.  For  how  much 
should  I  have  sold  it  to  gain  20  %  ? 


PERCENTAGE  167 

23.  A  merchant  engaged  in  business,  investing  cash  $  3500.  At  the 
end  of  one  year  he  found  that  he  had  paid  for  merchandise,  f  3250;  for 
rent,  $450;  for  clerk  hire,  $1200;  for  incidentals,  $725.  He  had  sold 
merchandise  to  the  value  of  $  6758.40.  What  was  his  per  cent  of  profit 
on  his  investment? 

24.  A  man's  real  estate  is  now  worth  $12,000.  The  first  year  it 
increased  20  %  in  value,  and  the  second  year  33^%.  What  did  it  cost  two 
years  ago? 

25.  A  dry  goods  merchant  marked  cloth  at  25%  advance  on  the  cost, 
but  was  obliged  to  sell  it  at  20%  less  than  the  marked  price.  If  it  cost 
him  $1  a  yard,  what  did  he  sell  it  for? 

26.  A  man  willed  50  7o  of  his  property  to  his  wife,  60%  of  the 
remainder  to  his  invalid  daughter,  and  the  remainder  to  his  church. 
The  church  received  $650.     What  did  the  wife  and  daughter  receive? 

27.  A  hardware  dealer  engaged  in  business  and  lost  12  %  of  his  money 
the  first  year  and  gained  33|%  the  second  year.  If  he  started  with 
$  4500,  how  much  has  he  now  ? 

28.  If  a  man  loses  $2400  by  selling  at  a  loss  of  12^%,  at  what  should 
he  sell  to  gain  12^%? 

29.  Mr.  McFarland  bought  a  square  piece  of  land,  containing  40 
acres,  paying  $800  for  it.  He  opened  a  street  through  the  center  of  it 
and  divided  the  land  on  each  side  into  lots  4  rods  wide.  If  he  sold  the 
corner  lots  for  $540  apiece,  and  the  other  lots  for  $400  apiece,  and  the 
expense  of  opening  up  the  land  was  $1250,  what  was  his  per  cent  of 
profit  or  loss? 

30.  A  speculator  bought  a  section  of  land  for  $  3  per  acre.  He  sold  J 
of  it  at  $8  per  acre,  and  the  remainder  at  $9.50  per  acre.  If  the  cost  of 
making  the  sales  was  $  135,  what  was  his  gain  per  cent  ? 

31.  A  merchant's  profit  the  second  year  was  66|%  greater  than  it  was 
the  first.  The  profits  of  the  two  years  amounted  to  $  8120.  What  was 
the  profit  of  each  year  ? 

32.  I  offered  my  house  and  lot  for  sale  at  50  %  above  cost,  but  sold 
them  for  25%  below  the  asking  price  and  gained  $600.  What  was  the 
cost  ?    What  the  gain  per  cent  ? 

33.  If  my  wheat  cost  90  j^  a  bushel  and  I  lose  10%  by  shrinkage, 
for  what  must  I  sell  it  to  make  a  net  gain  of  12 J  %  ? 

34.  If  I  sell  J  of  my  land  for  what  |  of  it  cost  me,  what  per  cent  do  I 
gain? 


168  ELEMENTS  OF  BUSINESS  ARITHMETIC 

35.  I  paid  $22,500  for  two  houses.  If  75%  of  the  cost  of  the  one  is 
equal  to  150  %  of  the  cost  of  the  other,  what  did  each  cost  ? 

36.  A  has  25%  more  money  than  B,  B  has  20%  more  than  C,  C  has 
12|  less  than  D.     How  much  has  each,  if  together  they  have  1 33,900  ? 

37.  I  paid  $580  for  a  horse,  wagon,  and  harness.  The  wagon  cost 
40  %  less,  and  the  harness  66|%  less,  than  the  horse.  What  was  the  cost 
of  each? 

38.  A  miller  in  3  years  made  gains  amounting  to  $6336.  The  second 
year's  gain  was  20%  greater  than  that  of  the  first,  and  the  third  10% 
greater  than  the  second.     What  was  each  year's  gains? 

39.  A  grocer  bought  apples  at  60^  a  bushel,  and  marked  them  so  as 
to  sell  at  a  gain  of  20%,  but  sold  them  at  a  reduction  of  12-|%  from  the 
marked  price.     If  he  gained  $42.80,  how  many  bushels  had  he? 

40.  A  man  sold  goods  that  cost  $425  at  an  advance  of  40%.  He  lost 
25  %  in  bad  debts  and  paid  5  %  for  collecting.  What  was  his  gain  or 
loss? 

41.  Our  stock  of  goods  decreased  in  value  33p/o,  and  again  20%.  It 
then  increased  20%,  and  again  33i%,  and  was  sold  at  a  loss  of  $66. 
What  was  it  worth  at  first  ? 

42.  If  the  retail  profit  is  33^%,  what  do  I  make  on  goods  that  cost 
$180,  if  I  sell  them  at  wholesale  for  10%  less  than  at  retail? 

43.  Last  year  a  merchant  gained  $2000.  This  year  he  gained  20% 
more,  which  is  44|%  of  what  he  gained  the  year  before  last.  What  did 
he  gain  each  year  ? 

44.  A  barrel  of  cider  had  lost  20%  by  leakage  and  was  sold  for  50% 
above  cost.     What  per  cent  gain  was  that  ? 

45.  I  buy  a  barrel  of  vinegar,  containing  52  gallons,  at  20  J^  a  gallon. 
If  four  gallons  leak  out,  for  what  must  I  sell  the  remainder  per  gallon  to 
gain  25%? 

46.  If  I  buy  stocks  for  80%  of  their  value,  and  sell  them  for  110%  of 
their  value,  what  per  cent  do  I  gain  ? 

47.  Sold  a  lot  of  cotton  at  a  gain  of  33^%.  With  the  money  I  bought 
another  lot  and  sold  it  for  $480  at  a  loss  of  20  %•  What  did  the  first  lot 
cost  me? 

48.  A  manufacturer  sold  at  a  profit  of  33|  %  to  a  wholesale  dealer, 
who  sold  at  a  profit  of  25%  to  the  retail  dealer.     The  retail  dealer  sold 


PERCENTAGE  169 

at  a  profit  of  20%  and  received  $60  for  the  article.    What  was  the 
original  cost? 

49.  A  merchant  increased  his  investment  50%.  He  then  withdrew 
66|%  of  his  capital  and  invested  it  in  bonds.  He  now  has  $4800  in  the 
business.     How  much  did  he  invest  at  first  ? 

50.  An  implement  man  bought  a  mower  for  $  42.  How  much  must 
he  ask  for  it  in  order  to  make  a  discount  of  25%  and  still  gain  16f  %? 

51.  I  buy  goods  and  sell  at  a  loss  of  10%.  I  reinvest  the  money  and 
gain  10%.     What  per  cent  do  I  gain  or  lose? 

52.  A  dairyman  sold  two  cows  for  $120,  and  gained  20%  on  the  one 
and  lost  20  %  on  the  other.  What  was  the  cost  of  each,  if  he  sold  the 
first  for  50  %  more  than  the  other  ? 

53.  A  real  estate  dealer  sold  a  building  for  $  15,000  and  lost  40%.  He 
sold  a  house  and  lot  at  the  same  time  and  gained  16f  7o-  He  did  not  gain 
or  lose  on  the  two  transactions.     What  was  the  cost  of  each  property? 

54.  A  man  bought  a  business  for  $9775,  which  was  15%  more  than 
the  former  owner  paid  for  it.  He  then  sold  it  at  a  profit  of  6%.  What 
was  the  selling  price  ?    What  the  gain  ?    What  the  original  cost  ? 

55.  The  imports  of  sugar  and  molasses  into  the  United  States  in 
1891  amounted  to  $108,387,388;  in  1900  they  amounted  to  $101,100,000. 
What  was  the  per  cent  of  decrease  ? 

56.  The  exports  of  wheat  and  flour  in  1891  and  1900  were  as  follows : 
1891,  $106,125,188;  1900,  $140,997,966.  What  was  the  per  cent  of 
increase  ? 

57.  The  world's  supply  of  sugar  in  1900  was  8,800,000  lb.  The  sup- 
ply of  cane  sugar  was  2,850,000  lb.  What  per  cent  of  the  total  supply 
was  cane  sugar  ? 

58.  The  exports  of  the  United  States  in  a  given  year  were :  Europe, 
$697,614,106;  Asia  and  Oceania,  $43,813,519 ;  British  North  American 
Possessions,  $37,345,515;  West  Indies,  $33,416,178;  South  America, 
$33,226,401;  Mexico  and  Central  America,  $21,236,545;  Africa, 
$4,738,847;  all  other,  $879,172.  Find  what  per  cent  of  the  total  was 
exported  to  each  country. 

59.  A  wholesale  dealer  had  sold  $1500  worth  of  goods  to  a  retailer. 
The  retail  dealer  failed  and  could  pay  only  75  f  on  the  dollar.  K  the 
wholesale  dealer  paid  5%  for  the  collection  of  ,the  debt,  what  was  his  per 
cent  of  loss  ? 


170 


ELEMENTS  OF  BUSINESS  ARITHMETIC 


60.  The  estimated  total  cut  of  lumber  in  the  United  States  from 
1880  to  1906  was  706,712,000  board  feet.  Of  this  amount,  Michigan 
produced  93,436,000.  Michigan's  output  was  what  per  cent  of  the 
total? 

61.  The  freight  moved  over  the  several  lines  of  railroad  in  the  United 
States  in  a  given  year  was  as  follows : 


Class  of  Commodity 

Tonnage  keported  as 

OKIGINATING   ON   LiNE 

Pek  Cent  op 
Aggkegatb 

Products  of  agriculture 

Products  of  animals 

Products  of  mines 

Products  of  forests 

Manufactures 

Merchandise 

Miscellaneous 

56,102,838 
15,145,297 
269,372,556 
60,844,933 
71,681,178 
21,697,693 
26,493,338 

? 
? 
? 
? 
? 
? 
? 

Grand  total 

9 

? 

Europe,  $215,000,000 


Find  the  aggregate  tonnage.     Find  the  per  cent  to  two  decimal  places 
that  each  item  is  of  the  aggregate. 

62.  If  the  exports  of  manu- 
factured products  from  the 
United  States  is  as  shown  in 
the  accompanying  square,  find 
what  per  cent  of  the  aggregate 
is  exported  to  each  division. 

63.  The  total  imports  into 
the  United  States  for  a  certain 
year  were  $903,320,948.  Of 
this  amount,  the  United  King- 
dom sent  18%;  Germany,  11%; 
France,  9%;  Brazil,  8.7%;  Brit- 
ish North  America,  5.3%;  all 
other  countries,  48%,.  Find  the 
amount  sent  by  each  division 
to  approximate  millions,  and 
Problem  62. 


North  America,  $96,000,000 


Asia,  $34,000,000 


Oceania,  $29,000,000 


South  America,  $27,000,000 


Africa,  $11,000,000 


represent  it  in   graphical  form   as  in 


PERCENTAGE  171 

64.  From  the  accompanying  graph,  showing  the  per  capita  consump- 
tion of  coffee  in  pounds,  find  the 

per  cent  of  the  whole  amount      United  Kingdom 72 

consumed  by   each  of  the  five      ~~ 

countries,  the  population  of  the      -^^^v ^° 

United  Kingdom  being  approxi-      Austria  Hungary 2.04 

mately     43,650,000;   of     Italy,      ^    ' 

33,750,000 ;  of  Austria-Hungary,      Germany 4.62 

47,000,000;      of      Germany,      

60,650,000;  and  of  the  United      United  States 10.79 

States,  90,000,000. 

65.  If  the  per  capita  consumption  of  sugar  in  Russia  is  13.9 ;  in 
Portugal,  14.2  ;  in  Austria,  17.6  ;  in  Belgium,  23.0 ;  in  Netherlands,  34.3 ; 
in  France,  36.9 ;  in  Norway  and  Sweden,  40.6 ;  in  Denmark,  48.7 ;  in 
Switzerland,  52.0;  in  the  United  States,  65.2;  in  the  United  Kingdom, 
91.1  pounds,  find  total  amount  consumed  by  all,  the  population  of  each 
being,  in  approximate  millions,  as  follows:  Russia,  107^;  Portugal,  5| ; 
Austria,  26;  Belgium,  6|;  Netherlands,  5|;  France,  39;  Norway  and 
Sweden,  7^;  Denmark,  2.6;  Switzerland,  3|;  United  States,  90;  and 
United  Kingdom,  43.6.  Find  the  per  cent  of  this  amount  which  each 
consumes.     Represent  in  graphical  form  as  in  Problem  64. 


XIV 

TRADE  DISCOUNT 

141.  Trade  Discount  is  an  allowance  from  the  price  of 
goods  made  because  of  special  conditions,  or  for  settlement 
of  the  account  within  a  specified  time.  Such  discounts  are 
usually  made  "to  the  trade"  or  to  persons  engaged  in  the 
retail  trade  in  a  given  line  of  goods,  hence  the  term.  The 
amount  of  the  allowance  is  expressed  in  per  cent  or  fractions. 
Discount  plays  a  large  part  in  the  arithmetic  of  mercantile 
business. 

142.  Marked  and  List  Prices.  Merchants  generally  mark 
or  list  their  goods,  so  they  may  gain  a  certain  per  cent  of  the 
cost  as  profit.  If  strictly  a  "  one  price  "  house,  the  per  cent 
of  profit  is  the  only  element  to  be  considered  by  a  retailer  in 
determining  the  "  marked  "  price.  If  th'e  goods  are  to  be  so 
marked  that  a  discount  may  be  allowed  and  still  a  certain 
per  cent  of  profit  be  made,  then  both  discount  and  profit 
must  be  considered. 

Wholesalers  and  manufacturers  very  generally  publish  a 
"  list  price  "  from  which  a  certain  per  cent  is  allowed  as  a 
discount  "to  the  trade."  Fluctuations  in  the  market  prices 
may  in  this  way  be  provided  for  by  varying  the  discounts 
quoted  instead  of  changing  the  printed  or  list  prices.  Goods 
listed  and  not  subject  to  discounts  of  any  kind  are  usually 
marked  "net." 

143.  Term  Discount.     Even  when  goods  are  billed  at  such 

172 


TRADE  DISCOUNT  173 

discounted  prices,  a  further  discount  is  often  allowed  if  the 
bttl  is  paid  within  a  certain  time,  say  30  or  60  days ;  while 
that  discounted  price  maybe  still  further  discounted  for  <?a«A. 

Thus,  goods  may  be  listed  at  $60,  and  quoted  at  dis- 
counts of  20,  10,  and  5.  The  goods  would  usually  be  billed 
at  "20  off"  (160  less  |),  or  148.  10%  off  for  payment 
within  a  time  period  marked,  say  30  days,  would  reduce  the 
bill  ($48-14.80)  to  $43.20,  and  5%  further  discount  for 
cash  would  make  the  cash  value  of  the  bill  J  of  $4.32 
($2.16)  less,  or  $41.04. 

When  discounts  are  quoted  in  series  as  above,  it  should  be 
.noted  that  the  discount  indicated  by  the  first  per  cent  given, 
is  taken  from  the  amount  of  the  bill,  then  the  second  dis- 
count is  computed  upon  the  remainder  and  deducted,  and 
so  for  the  third,  each  discount  being  a  per  cent  of  the 
remainder  after  deducting  the  previous  discount. 

144.  Fractional  Discounts  Used.  Whenever  the  nature  of 
the  business  will  allow,  prices  are  listed  so  that  the  discounts 
used  are  those  per  cents  which  are  the  equivalents  of  simple 
fractions.  Thus,  instead  of  allowing  a  discount  of  15%,  the 
list  price  is  raised  slightly  so  that  a  discount  of  either  16|  % 
or  20  %  may  be  given.  Finding  |  is  not  only  purely  a  mental 
operation,  but  the  process  is  simpler,  shorter,  and  less  liable  to 
error  than  multiplying  y^-g-  of  the  amount  by  15.  For  these 
reasons,  probably,  the  trend  of  business  usage  for  years  past 
has  been  away  from  a  purely  decimal  to  a  fractional  discount. 

145.  Finding  Net  Price.  1.  200  arithmetics  were  billed  to 
a  dealer  for  $  80,  on  which  a  discount  of  20  %  was  allowed. 
What  was  the  net  cost  ? 

20%  =i 
I  of  $80  =  $16 
$80 -$16  =  $64,  net  cost. 


174  ELEMENTS  OF  BUSINESS  ARITHMETIC 

2.  A  discount  of  $25  was  allowed  on  a  bill  of  goods  for 
$150.     What  was  the  per  cent  or  rate  of  discount  ? 

$25  =  ^2^^  of  1150. 

i¥iT  =  6  ^^  l^t  %^  ^^t®  of  discount. 

146.  Net  Price  with  a  Discount  Series.  1.  Discounts  of  25, 
20,  and  5  per  cent  were  quoted  on  a  bill  of  goods  amounting 
to  $360.     What  was  the  net  price? 


$360. 

=  the  list  price. 

90. 

=  25% 

or  I  of  $360. 

270. 

=  price 

after  first  discount. 

54. 

=  20% 

or  1  of  $270. 

216. 

=  price 

after  second  discount. 

10.80 

=  5%  or  2V  of  $216. 

$205.20 

=  net  price. 

2.    A  bill  of  goods  netted  $  216,  after  being  discounted  at 
25 %  and  20 %.     What  was  the  face  of  the  bill? 

$216  =  20  %  or  ^  less  than  the  amount  of  the  last  discount. 

I  =  $216. 

|  =  f  of  $216,  or  $270. 
$270  =  25  %  or  ^  less  than  face  of  bill. 

f  =  $270. 

I  =  f  of  $270  or  $360,  face  of  bill. 

147.   Single  Equivalent  of  Discount  Series.     1.    Discounts 
of  25%  and  20%  are  equivalent  to  what  single  discount? 

100  %  =  the  whole  of  anything. 

25  %  =  J  of  100  %,  or  first  discount. 

75  %  =  price  after  first  discount. 

15%  =  20%,  or^of  75%. 

60  %  =  net  price  after  second  discount. 
100  %  =  60  %  =  40  %,  equivalent  single  discount. 


TRADE  DISCOUNT  175 

2.    A  bill  of  goods  netted  $216,  after  being  discounted  25 
and  20  per  cent.     What  was  the  face  of  the  bill  ? 

Discounts  of  25  and  20  =  40  %  discount  (Prob.  1). 
100  %  -40  %  =  60  %,  net  price. 
60%,  or  I  =  1216. 

f  =  f  of  1216,  or  1360. 

PROBLEMS 

1.  A  merchant  bought  a  bill  of  goods  amounting  to  $400.  He  re* 
ceived  a  discount  of  10  %  and  8^  %  off  for  cash.  What  was  the  cash 
value  of  the  bill  ? 

2.  Fonts  &  Co.  bought  a  bill  of  goods  amounting  to  $600,  on  which 
they  received  a  discount  of  20  %  and  10  %  off  for  cash.  What  was  the 
cash  value  of  the  bill  ? 

3.  A  carriage  is  marked  at  $100,  but  the  dealer  tells  me  I  may  have 
it  on  30  days'  time  at  a  discount  of  20  %,  and,  if  I  pay  cash,  an  additional 
discount  of  2|  %  will  be  given.     What  must  I  pay  for  it  in  cash? 

4.  The  marked  price  of  a  mower  is  %  64.  I  sell  it  to  a  farmer,  giving 
him  a  discount  of  12|  %  and  14|  %.     What  did  he  pay  ? 

5.  I  buy  a  bill  of  goods  amounting  to  %  500,  on  which  I  get  a  dis- 
count of  12|  %  and  4  %  off  for  cash.     What  did  I  pay  ? 

6.  Smith  &  Co.  buy  a  bill  of  goods  amounting  to  $  500,  on  which 
they  get  a  discount  of  16|%  and  10%  off  for  cash.  What  did  they 
pay? 

7.  McKee  &  Co.  sell  goods  for  %  120.  They  had  given  a  discount  of 
20  %  and  5  %  off  for  cash.     What  was  the  list  price  of  the  bill  ? 

8.  What  is  the  cash  value  of  a  bill  of  goods  that  lists  $49,  if  a  dis- 
count of  14f  %  and  16|%  is  given? 

9.  Mr.  Kerr  sells  a  wagon  on  60  days'  lime  for  $72.  If  paid  in 
30  days,  the  buyer  will  get  a  discount  of  16|%;  if  he  pays  in  10 
days,  he  will  get  10  %  additional.  What  would  the  "  cash  in  10  days  " 
payment  be? 

10.   A  dealer  bought  goods  at  20%  and  25  %  off,  and  sold  them  at  10% 
and  10  %  off  the  same  marked  price.     What  was  his  per  cent  of  gain  ? 


176 


ELEMENTS   OF  BUSINESS  ARITHMETIC 


11.  How  much  better  is  a  series  of  25%,  20%,  and  10%  than  a  single 
discount  of  60%? 

12.  What  is  the  amount  of  the  following  bill  if  it  is  paid  within  10 
days? 


M  ^^^^.yU 


Chicago,  DL,. 


fyP-^^^-<^.A^^^ 


«'  W.  M.  WELCH  &  COMPANY 

Tenns:   ^ , ./^^.^ ^J,  .9 -/.  / ^  dr,^  "^  "«"«^  Street 


13.  Sold  Lew  A.  Wallace,  terms  5/10-n/30 ;  12  bx.  Apricots  @  %  2.15 ; 
12  bx.  Layer  Raisins  @  $2.95;  5  bx.  Dried  Citron  @  $1.33^;  3  tierces 
Refined  Lard,  1065 #,  @  9^;  3  cases  Shredded  Codfish,  6  doz.,  @  |1.12^; 
4  cases  Salmon,  8  doz.,  @  f  2.12|.     Find  cash  value. 

14.  Sold  A.  O.  Bennett,  terms  3/10-n/60;  4  cases  Rio  Coffee,  400  #, 
@Vl\f',2  bags  Mocha  Coffee,  185 #,  @  28^  ;  2  bags  M.  G.  Coffee,  378  #, 
@  22  ^ ;  4  bbl.  Herring  @  % 6.25.     What  is  the  cash  value  of  the  bill? 

15.  Sold  W.  B.  Lambert,  terms  10-2/10-n/30;  5  doz.  Baseballs 
@$15;  3  doz.  Bats  @$9;  24  Infielders  Gloves  @  $2.50;  3  Catcher's 
Gloves  @  $6.25;  12  Baseball  Uniforms  @  $14;  12  pr.  Shoes  @  $3.50. 
What  is  the  cash  value  of  the  bill  ? 

16.  Bought  of  J.  A.  Amsbaugh  &  Co.,  225  bbl.  Flour  @  $4.45;  165 
S.  H.  Hams,  1960  #,  ©14^^;  5  tierces  Refined  Lard,  745#,  @  9^2^;  6  bbl. 
Mess  Pork  @  $15.20;  8  bbl.  G.  Flour  @  $6.10;  24  bx.  Figs  @  $1.85; 
5' cases  G.  Coffee,  300  #,  @  $0.12|.  I  am  allowed  a  discount  of  20%  and 
an  additional  discount  for  cash  of  5  %.  What  is  the  cash  value  of  the 
bill? 

17.  A  man  desires  to  buy  48,000  ft.  of  pine  lumber.  One  firm  offers 
the  lumber  at  $60  per  M,  less  discounts  of  20%  and  5%;  another  firm 


tra.de  discount  177 

offers  the  lumber  at  $75  per  M,  less  discounts  of  33^%  and  6|%.  The 
terms  offered  by  both  are  l/lO-n/30.  Which  is  the  better  offer,  and 
how  much  does  the  lumber  cost? 

148.  Marking  Goods  for  the  Shelf.  In  determining  the 
selling  price  of  goods,  it  is  usual  to  add  to  the  cost  a  certain 
per  cent  as  profit.  Thus,  if  a  set  of  books,  costing  83.60,  is 
to  be  sold  at  a  profit  of  25  %,  they  must  be  marked  at  ^  more 
than  they  cost,  or  14.50. 

In  marking  goods  for  sale,  merchants  often  place  the  cost 
price  as  well  as  the  selling  price  on  each  article.  For  this 
cost  price  a  "  cipher  "  is  often  used.  Any  word  or  phrase  of 
ten  letters,  or  any  ten  arbitrary  characters,  may  be  used  as  a 
cipher.     Thus, 


Repeaters  x  and  y. 


Repeaters   #   -/- 

When  a  figure  occurs  twice  or  more  in  succession,  a  re- 
peater is  used  instead  of  repeating  the  character  for  the  fig- 
ure.    This  renders  deciphering  more  difficult. 

The  cost  and  selling  price  may  both  be  written  on  the 
tag.  If  so,  the  selling  price  is  written  below  a  horizontal 
line,   and    may   be    in    cipher    or    plainly       |_r|v  1.40 

written.  LHT  1-75 

Sometimes  the  cost  price  is  written  in  an 
entirely  different  cipher,  known  only  to  the       mon  2.48 

buyer  or  proprietor.  pmr  3.25 

149.  Listing  Goods  for  Catalogues.  Goods  to  be  listed  in  a 
catalogue  are  usually  so  priced  as  to  permit  a  discount  or  a 
discount  series,  and  still  to  sell  at  a  price  which  will  afford 
the  desired  per  cent  of  profit. 


1     2 

3 

4    5     6     7     8     9 

0 

i    m 

P 

or     t     a     n     c 
or 

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J     X 

178  ELEMENTS  OF  BUSINESS  ARITHMETIC 

Thus,  if  the  set  of  books  (Sec.  148)  costing  $3.60  is  to  be 
so  listed  that  a  discount  from  the  list  of  16|  %  may  be  given 
and  still  yield  a  profit  of  25  %,  the  selling  price  of  $4.50 
(Sec.  186)  is  J  less  than  the  list  price,  or  |  of  list.  The  list 
price,  then,  would  be  14.50  x  |,  or  15.40. 

If  it  were  desired  so  to  list  an  article,  costing  $  3.92,  that  a 
discount  of  25%,  16|%,  and  2%  might  be  given,  and  still 
provide  for  a  profit  of  25%,  the  steps  in  the  process  would 
be  as  follows : 

I  of  13.92  =  1 4.90,  selling  price. 

$  4.90  =  98  %  or  |f  of  price  discounted  2  %. 
$4.90  X  |^  =  $5.00. 

$5.00  =  f  of  price  discounted  16|  %. 
$5.00  X  f  =  $6.00. 

$  6.00  =  75  %  or  f  of  list  price.  . 
$6.00  X  I  =  $  8.00,  list  price. 

PROBLEMS 

1.  I  buy  goods  for  $24  and  desire  to  sell  them  so  I  may  make  33|%. 
What  must  I  ask  for  them  that  I  may  give  a  discount  of  20%  and  still 
make  my  desired  gain  ? 

2.  What  must  goods  that  cost  20  ^  a  yard  be  marked  so  as  to  make 
25  %  after  a  discount  of  16|  %  ? 

3.  What  must  I  ask  for  goods  for  which  I  paid  $  16,  if  I  desire  to 
give  a  discount  of  20%  and  still  gain  25%? 

4.  A  book  seller  bought  a  set  of  encyclopedias  for  $40.  Remarked 
them  so  as  to  gain  10  %  and  give  a  discount  of  20  %.  What  did  he  ask  ? 
What  did  he  get? 

5.  A  hardware  merchant  buys  stoves  for  $24  each.  He  desires  to 
sell  them  so  as  to  make  12|%  after  deducting  25 7o-  What  did  he  ask? 
What  did  he  get? 

6.  I  wish  to  sell  goods  at  cost  that  cost  me  $400.  In  order  to  do  so, 
I  mark  them  up  25%.     What  per  cent  discount  can  I  give  ? 

7.  I  buy  plows  at  $  9.60.  If  I  wish  to  make  a  gain  of  25  %  and  still 
offer  a  discount  of  25%  and  10%,  what  must  I  ask  for  them? 


TRADE  DISCOUNT  179 

8.  A  dealer  marked  a  carriage  by  mistake  at  $  160,  and  sold  it  at  a 
discount  of  25  %,  thereby  losing  16|  %.  What  should  he  have  marked  it 
in  order  to  have  gained  25%? 

9.  Wishing  to  sell  goods  at  cost  that  cost  me  $5, 1  mark  them  at 
20%  advance.     What  discount  must  I  give? 

10.  A  merchant  buys  goods  for  $  1200.     What  must  he  ask  in  order 
to  give  a  discount  of  16|%  and  still  gain  10%? 

11.  How  must  goods  that  cost  25  ^  be  marked  so  as  to  make  20  %  and 
still  give  a  discount  of  16|  %? 

12.  Find  the  list  price,  if  goods  that  cost  $  600  are  sold  at  a  gain  of 
25  %  after  a  discount  of  20  %,  5  %,  and  2  %. 

Using  the  characters  in  Sec.  148,  mark  the  cost  and  selling  price  of 
the  following  articles : 

13.  Shoes  costing  $5  and  selling  for  $  6.50. 

14.  Gloves  costing  $  22.50  per  dozen  and  selling  at  20%  gain. 

15.  Caps  costing  $  7.25  per  dozen  and  selling  at  33^ %,  gain. 

16.  Shoes  costing  1 1.97  and  selling  at  25  %  gain. 

17.  Boots  costing  $2.68  and  selling  at  $3.75. 

18.  Make  a  key  from  the  letters  contained  in  the  words  "blacksmith" 
and  "  authorizes,"  and  mark  the  articles  given  in  the  above  examples. 


XV 

COMMISSION 

150.  The  Commission  Business.  In  the  development  of 
our  industries,  certain  cities  become  the  centers  for  han- 
dling particular  products.  Thus,  Chicago  is  a  great  grain 
market;  and  Chicago,  Kansas  City,  and  Omaha  are  great  live- 
stock markets. 

Where  important  products  are  handled  so  largely,  there 
will  naturally  be  a  large  number  of  buyers,  and  sellers  will 
ship  their  products  from  the  tributary  territory.  It  would 
be  too  expensive  to  accompany  their  shipments  to  market, 
and  yet  they  want  their  goods  sold  to  the  highest  bidder. 
They  must,  then,  have  some  one  to  represent  them.  This 
representative  or  agent  of  the  owner  of  the  product  is 
known  as  a  commission  merchant. 

These  merchants  receive  the  goods,  see  that  they  are 
properly  cared  for,  effect  a  sale,  and  return  the  proceeds  to 
the  owner,  less  expenses  in  handling  and  a  fee  for  the  serv- 
ices rendered.  This  fee  is  usually  computed  as  a  per  cent 
of  the  amount  of  the  sale,  and  is  called  a  commission. 

151.  Terms  Used.  Goods  sent  to  an  agent  for  sale  are 
called  shipments^  and  are  received  by  the  agent  or  consignee 
as  a  consignment^  the  title  to  the  goods  remaining  with  the 
consignor.  The  statement  of  the  goods  received,  their  sales 
in  detail,  with  the  expenses  and  charges  thereon,  showing 
the  net  proceeds,  is  called  an  account  sales. 

In   a  like   manner   commission   merchants   often   receive 

180 


COMMISSION  181 

orders  to  buy,  as  agents,  certain  goods.  For  this  service 
they  are  paid  a  commission.  A  detailed  statement  of  the 
purchases,  with  expenses,  charges,  etc.,  is  called  an  account 
purchase, 

152.  Commission  in  General.  In  general  any  percentage 
of  money  or  value  handled,  which  is  received  by  an  agent 
for  his  services  as  such  agent,  is  a  commission.  Thus,  a  real 
estate  agent  receives  a  commission  on  property  sold;  an 
insurance  agent  a  commission  on  insurance  written ;  a  col- 
lector, a  commission  on  money  collected,  etc. 

153.  Commission  on  Purchases.  Sometimes,  an  amount  of 
money  is  sent  an  agent  witli»  instructions  to  expend  it  in  the 
purchase  of  goods.  The  present  tendency  in  business  is  to 
charge  commission  on  the  whole  amount  sent  (gross).  But 
many  commission  houses  still  charge  only  on  the  purchase 
(net).  When  the  amount  sent  is  to  include  the  sum  used 
for  the  purchase,  together  with  the  commission,  it  is  equal 
to  100  cfo  of  the  purchase  to  be  made,  plus  the  rate  to  be 
paid  as  commission  on  such  purchase. 

Thus,  if  1367.50  were  sent  to  a  commission  merchant 
with  which  to  buy  wheat,  after  deducting  his  commission 
of  5%,  what  would  he  invest  in  wheat? 

The  amount  of  money  he  received  would  include  100  %  of 
the  value  of  the  wheat  purchased  and  5  %  of  such  purchase  as 
commission.  $367.50  would  then  equal  105  %  of  the  amount 
to  be  expended  on  wheat.  105  %  =  f^.  Then  $367.50  =  |^ 
of  the  wheat  purchase  money.  gV  would  be  2^  of  $367.50, 
or  $17.50,  the  commission,  and  |^  would  be  $350,  or  the  money 
for  the  purchase  of  wheat. 


182 


ELEMENTS  OF  BUSINESS  ARITHMETIC 


PROBLEMS 

1.  How  many  pounds  of  coffee,  at  27^  per  pound,  can  be  bought  for 
$8424,  if  the  agent  is  allowed  4%  of  purchase  price  for  buying? 

2.  I  remitted  $2612.90  to  a  New  York  agent  for  the  purchase  of 
gloves.  If  the  agent's  commission  is  4%  net,  and  he  makes  an  added 
charge  of  1  %  for  guaranteeing  quality  of  goods,  how  many  dozen  pairs  of 
gloves,  at  $8.50  per  dozen,  should  he  send  me? 

3.  I  remitted  $600  to  an  agent  for  the  purchase  of  peaches.  If  the 
agent's  charges  were  5%  net  for  purchase  and  $12  for  inspection,  how 
many  baskets  did  he  buy  at  43/? 

4.  Rule  a  sheet  of  paper,  copy  the  following  account  sales,  making 

all  necessary  extensions,  etc. 


diicago,  UL, 


uf^y^ 


^(?^  ,.,.,19. 


Sold  for  the  account  of 


By  HOWARD  PAYNE  &  COMPANY 

COMMISSION  MERCHANTS 
159  Randolph  Street 


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5.  Prepare  an  account  sales  under  date  of  May  15,  1910  for  6000  bu. 
Wheat,  3000  #  Beef,  sold  Westerfield  Bros.,  Greenville,  O.,  for  the 
account  of  W.  C.  Pierse  &  Co.,  Union  City,  Ind.,  Sales:  April  10,  3000 
bu.  Wheat  @62^j^,  1500 #  Beef  @  9^^;  May  12,  3000  bu.  Wheat  @65^, 
1500 #  Beef  @10j^.  Charges:  freight,  $125;  cartage,  $15;  storage, 
$17.50;  insurance,  \%\  commission,  2%. 


COMMISSION 


183 


6.   Rule  a  sheet  of  paper  and  copy  the  following  purchase,  making 
all  extensions,  etc. 


New  York,  N.Y., 


6^-^-/  Zff        10 


Purchased  by 

L.  M.  BARKER  CS,  CO. 

For  the  account  and  risk  of 


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O'v^    yltyp^ 


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Amount  charged  to  your  account 


7.  Prepare  an  account  purchase  for  coffee  purchased  by  E.  H.  Reed 
&  Co.,  June  15,  for  the  account  and  risk  of  Ames,  Spencer  &  Co.  Pur- 
chases: 8  mats  G.  P.  Coffee,  600  #  at  2Zf\  16  mats  M.  Coffee,  1300  #  at 
34^;  24  mats  J.  Coffee,  1800 #  at  30^.  Charges:  cartage,  |6.70;  com- 
mission, 2  %  net. 

8.  A  man  places  a  claim  of  $2500  in  the  hands  of  an  attorney  for 
collection.  If  the  debtor  is  a  bankrupt,  having  liabilities  to  the  amount 
of  $20,000  and  resources  to  the  amount  of  $15,000,  how  much  should 
the  creditor  receive  after  a  commission  of  2^%  is  deducted  for  collection  ? 

9.  A  principal  sends  his  agent  $2500  with  which  to  buy  corn.  After 
deducting  his  commission  of  2i%  on  the  amount  sent,  he  buys  corn,  pay- 
ing 45^  a  bushel.     How  many  bushels  does  he  buy? 

10.  Find  the  rate  of  commission  and  guaranty  on  a  purchase  if  the 
total  cost  is  $4,143.10,  the  commission  $174.60,  the  cartage  $40,  and 
guaranty  $48.50. 

11.  A  land  agent  receives  $51,000  with  which  to  buy  land.  After 
deducting  his  commission  of  5%  on  the  amount  sent,  how  many  acres 
can  he  buy  at  $35  per  acre? 


184         ELEMENTS  OF  BUSINESS  ARITHMETIC 

12.  I  sent  my  agent  $4080  with  which  to  buy  corn.  After  deducting 
his  commission  of  2%  net,  how  many  bushels  can  he  buy  at  54 j*  a 
bushel? 

13.  A  dairyman  received  from  his  city  agent  $980  as  the  net  proceeds 
of  a  shipment  of  butter.  If  the  agent's  commission  is  2^%,  delivery 
charges  $12.55,  how  many  pounds  at  25 f^  a,  pound  must  he  have  sold? 
What  was  the  agent's  commission? 

14.  An  agent  remits  to  his  principal  $3656.25  after  deducting  a  com- 
mission of  $93.75.  What  was  the  selling  price  of  the  goods?  What 
was  the  rate  of  commission  ? 

15.  An  agent  sells  cotton  for  $3600  and  charges  2%  commission. 
After  deducting  his  commission  of  3%  for  buying,  he  buys  corn  at  30^ 
a  bushel.     How  many  bushels  does  he  buy  ? 

16.  Find  the  proceeds  of  a  sale,  if  the  agent  charges  2|%  commission, 
$17.50  for  cartage,  $10.45  for  storage,  and  $4.25  for  insurance.  The 
net  proceeds  are  $1545.74. 

17.  I  sell  goods  for  $3600  and  charge  3%  commission.  I  then  invest 
the  proceeds,  less  a  commission  of  3%  on  the  proceeds,  in  grain.  How 
much  do  I  invest  in  grain  ? 

18.  An  agent  charged  me  5%  for  selling  wheat,  and  2%  for  investing 
the  proceeds  in  cotton.  His  commission  was  $558.90;  what  was  the 
selling  price  of  the  wheat? 

19.  I  sent  my  agent  in  Missouri  $2000  with  which  to  buy  apples  at 
$1.75  per  barrel  and  to  pay  commission  and  drayage.  His  charges  were  : 
commission  4%,  drayage  5^  per  barrel.  How  many  barrels  did  he  buy, 
and  what  was  his  unexpended  balance? 

20.  Render  in  full  the  following  account  sales,  supplying  rates  for 
insurance  and  commission :  April  16,  Jacob  Speer  &  Co.,  Portland,  Ore., 
sold  Henry  Williams  &  Co.,  Chicago,  12,000  Salmon  at  11^;  18,000 
Halibut  at  16^.  Charges:  freight,  $250;  insurance,  $32;  commission, 
$128. 

21.  Render  an  account  sales  for  the  following  sales,  supplying  all 
rates:  The  Henry  St.  Clair  Co.,  Cleveland,  O.,  sold  for  A.  W.  Ray- 
mond, Sandusky,  O.,  250  bu.  Beans  at  $1.06;  400  lb.  Cheese  at  10^;  40 
bbl.  Flour  at  $  5.60 ;  1500  bu.  Wheat  at  65  ft.  Charges :  freight,  $  84.60 ; 
storage,  $24.30;  inspection,  $4;  insurance,  $3.76;  commission,  $75.20. 


XVI 

TAXES  AND  DUTIES 
TAXES 

154.  Property  Tax.  Money  levied  by  the  government  for 
defraying  its  expenses  is  called  a  tax.  For  local,  county,  and 
state  purposes  this  tax  is  usually  levied  upon  property. 

An  assessor  lists  all  the  property  in  his  district  and  places 
a  valuation  upon  it.  This  valuation  is  either  an  estimated 
actual  cash  value  or  a  certain  per  cent  of  it,  as  may  be  deter- 
mined by  law  or  custom  in  the  locality.  There  is  usually 
a  board  of  review  or  equalization  in  the  local  district,  county, 
and  state,  which  compares  valuations  of  different  classes  of 
property  made  by  the  different  assessors,  and  equalizes  them. 
The  total  of  all  the  property  valuations  constitute  the  assessed 
valuation  of  the  township,  village,  school  district,  county,  or 
state.  An  estimate  is  then  made  of  the  amount  needed  for 
the  governmental  expenses  of  each  of  such  units,  and  the 
per  cent  such  an  amount  is  of  the  assessed  valuation  is  the 
tax  rate  levied.  This  per  cent  is  so  often  fractional  that  it  is 
usually  expressed  in  mills ;  meaning  the  number  of  mills 
paid  as  tax  on  each  dollar  of  property  valuation. 

155.  Kinds  of  Property.  Property  is  of  two  kinds — real 
estate^  or  lands  with  their  improvements  and  buildings,  and 
personal  property/,  which  includes  all  other  property.  Assess- 
ments are  made  separately  for  each  kind  of  property,  and 
often  the  personal  and  real  taxes  are  due  and  become  delin- 
quent at  different  dates.  Where  taxes  become  delinquent, 
a  penalty  is  usually  charged  for  their  nonpayment  on  time. 

185 


186  ELEMENTS  OF  BUSINESS  ARITHMETIC 

156.  Other  Forms  of  Taxation.  Taxes  levied  solely  upon 
tangible  property  often  lead  to  serious  inequalities.  Efforts 
to  distribute  the  burdens  of  government  according  to  the 
benefits  derived  therefrom  and  according  to  the  ability 
to  pay,  have  led  to  many  forms  of  special  taxation,  such  as 
taxes  upon  capital  stock,  gross  earnings,  or  mileage  of  wires 
or  railroad  bed,  etc.  Franchises^  granted  by  local  legislative 
bodies,  giving  to  private  corporations  the  right  to  use  the 
streets  for  pipes,  rails,  or  wires,  etc.,  are  often  taxed.  Inheri- 
tances and  incomes  are  also  taxed  in  some  countries  and  states. 
Revenue  is  derived,  too,  from  the  sale  of  licenses^  from  fees 
for  official  acts,  and  from  fines  imposed  for  breaches  of  law. 
In  some  states  a  tax  is  laid  upon  each  voter,  regardless  of 
property  owned.  This  is  a  per  capita  tax,  and  it  is  known  as 
a  poll  tax. 

157.  Finding  the  Rate  of  Taxation.  Example.  A  township 
board  finds  that  the  assessed  valuation  of  all  property  in  the 
township  is  $600,000,  and  the  total  amount  of  revenue  needed 
is  $15,000.  How  much  tax  must  be  levied  on  each  dollar  of 
property  valuation  ? 

If  115,000  must  be  raised  on  a  valuation  of  $600,000,  the 
part  taken  for  taxes  will  be  sYAVtT'  ^^  lo'  which  is  2|-%,  or 
25  mills,  on  the  dollar. 

158.  Finding  the  Tax.  Example.  A  man's  assessment  is 
$16,000.  The  total  rate  of  taxation  for  all  purposes  in  the 
city  of  his  residence  was  25  mills.  What  was  his  individual 
tax? 

If  he  was  to  pay  25  mills  on  each  dollar,  he  would  pay  2  J  %, 
or  ^1^,  of  his  property  value  as  a  tax. 
^-^  of  $16,000  is  $400,  amount  of  tax. 

PROBLEMS 

1.  If  property  is  valued  at  $  500  and  the  taxes  are  $  5,  what  is  the 
rate  of  taxation  ? 


TAXES  AND  DUTIES  187 

2.  If  property  is  valued  at  1 25,000  and  a  tax  of  $500  is  to  be  paid  on 

it,  what  is  the  rate? 

3.  The  property  in  a  town  is  valued  at  $986,600  and  the  amount  of 
revenue  to  be  raised  |4933.  What  would  be  my  tax  if  my  property  is 
valued  at  $5500? 

4.  The  assessed  valuation  of  a  man's  property  is  $12,500,  and  the 
rate  of  taxation  is  23  mills  on  the  dollar.     What  is  his  tax  ? 

5.  The  real  estate  of  a  city  is  valued  at  $  9,864,000.    A  revenue  of    i\  p  ^ 
$108,640  must  be  raised.     The  tax   on  personal  property  is  $12,000.    ^ 
What  will  be  the  rate  on  real  estate  ? 

6.  What  will  be  a  man's  tax,  at  the  above  rate,  who  has  property     ; 
valued  at  $6400? 

7.  The  value  of  the  property  in  a  township  is  $  236,600  and  the  amount 
of  money  needed  for  state  purposes  is  $709.80.  What  is  the  rate  of 
taxation  for  the  state  ?  The  rate  fixed  by  the  school  board  is  9.4  mills. 
What  amount  is  collected  for  school  purposes  ? 

8.  I  have  property  valued  at  $  9800  and  the  total  rate  of  taxation  is  i/ 

6.4  mills.     How  much  tax  do  I  pay  ? 

9.  The  personal  property  in  a  certain  township  is  valued  at  $  120,000 
and  the  real  estate  at  $2,400,000.  The  personal  property  is  taxed  at  f 
its  value  and  the  real  estate  at  I  its  value.  If  the  rate  is  25.8  mills, 
how  much  tax  is  collected  ? 

-'  10.  The  expenses  of  a  certain  town  are  $595,512.80.  The  tax  rate 
is  16  mills  on  the  dollar.     What  is  the  assessor's  valuation  ? 

11.  The  assessed  valuation  of  a  county  as  fixed  by  the  board  of 
equalization  is  $25,395,180.      The  rates  of  taxation  are  as  follows  :  State, 

3.5  mills  ;  State  University,  2  mills;  State  Agricultural  College,  1.5  mills; 
State  Normal  School,  1  mill ;  County,  3  mills ;  County  School,  1  mill ; 
County  Bridge,  3  mills;  Insane  Hospital,  .5  mill;  County  Bond,  2  mills; 
County  Poor,  .2  mill;  Soldier's  Relief,  .3  mill;  County  Road,  2  mills; 
Board  of  Health,  .1  mill;  Teacher's  Fund,  9.4  mills  (average)  ;  School 
Contingent  Fund,  3.7  mills  (average).  What  is  the  total  tax  raised  in 
the  county  ?  What  is  the  total  rate  of  taxation  ?  What  is  property  taxed 
that  is  valued  at  $5400?  What  is  the  amount  of  tax  levied  for  each 
purpose? 

12.  A  tax  of  $  6649.60  is  to  be  raised  in  a  certain  city.  The  valuation 
of  the  real  property  is  $898,500,  which  is  assessed  at  f  its  valuation. 
The  personal  property  is  assessed  at  $65,000.     There  are  1200  polls, 


188  ELEMENTS  OF  BUSINESS  ARITHMETIC 

assessed  at  82  each.  What  is  the  rate  of  taxation,  the  rate  for  collection 
being  1%!  rWhat  is  a  man's  tax  whose  real  estate  is  valued  at  $  14,500, 
and  personal  property  at  $  l^OO  ?  ^ 

DUTIES 

159.  National  Taxation.  National  revenue  is  raised  largely 
by  the  forms  of  taxation  known  as  duties  and  excises.  Im- 
port duties  are  taxes  levied  upon  goods  imported  into  a 
nation.  In  the  United  States,  Congress  establishes  a  tariff 
or  list  of  such  duties.  These  duties  are  collected  at  the 
custom  houses  to  be  found  at  the  various  ports  of  entry.  The 
officer  in  charge  of  the  custom  house  is  called  a  collector. 

Import  duties  are  of  two  kinds  :  those  consisting  of  fixed 
charges  on  certain  units  of  measurements  of  commodities,  as 
the  gallon,  pound,  or  yard,  called  specific  duties,  and  those 
which  form  a  certain  per  cent  of  the  value  of  the  goods  be- 
ing imported,  called  ad  valorem.  Specific  duties  only  are 
levied  on  certain  articles,  ad  valorem  only  on  others,  and  on 
some,  duties  of  both  kinds  are  levied.  A  statement  or  in- 
voice, showing  the  various  articles  being  imported,  the  dis- 
tinguishing mark,  number,  or  quantity  and  value  of  each,  is 
termed  a  customs  declaration  or  manifest. 

When  the  goods  are  purchased  in  a  foreign  country  for 
import,  an  invoice  giving  descriptions  and  prices  is  filed 
with  the  United  States  Consul  in  the  district  where  the  pur- 
chase is  made,  and  he  sends  a  copy  to  the  collector  of  the 
port  to  which  the  goods  are  shipped.  When  the  goods 
arrive,  the  cases  are  opened  and  the  quantity  and  value  as 
compared  to  that  stated  by  the  invoice  is  ascertained.  For 
too  great  an  undervaluation  a  fine  may  be  imposed  or  the 
goods  confiscated.  Travelers'  baggage  is  also  inspected  for 
dutiable  goods. 

Instead  of  paying  duty  at  once,  goods  may  be  stored  in 
a  bonded  warehouse,  from  which  they  may  be  withdrawn  for 


TAXES  AND   DUTIES 


189 


export  or  upon  payment  of  the  duty.  Such  goods  are  said 
to  be  in  Bond.  Importing  goods  without  the  knowledge  of 
the  government  officers,  thus  avoiding  the  payment  of 
duty,  is  called  smuggling.,  for  which  heavy  penalties  are 
prescribed. 

Excises  are  taxes  levied  upon  certain  specified  articles,  as 
liquors,  tobacco,  opium,  oleomargarine,  etc.,  which  are  grown 
or  manufactured  in  this  country.  For  the  collection  of  these 
taxes,  there  are  National  Revenue  districts,  and  collectors  for 
each  district. 

Table  of  Custom  Duties  on  Certain  Articles 


Abticles 


Tariff  Eate 


Alcohol 

Brushes    

Carpets,  treble  ingrain 

Carpets,  two-ply 

Carpets,  tapestry,  Brussels 

Carpets,  Wilton,  Axminster 

Cigars  and  cigarettes 

Cotton  hosiery  (made  on  machines  and  not 

otherwise  provided  for) 

Cotton  plushes,  unbleached 

Cutlery,  more  than  $3  per  doz 

Cutlery,  razors,  $3  or  more  per  doz.       .     . 

Cutlery,  sword  blades 

Dyewoods,  common  extracts  of  .... 
Glass,  polished  plate  not  over  384  sq.  in.     . 

Gloves,  men's  kid 

Glue,  value  not  over  10  j^  per  lb 

Hay 

Linen,  wearing  apparel 

Silk,  woven       

Silk  laces,  wearing  apparel 

Skins,  uncured,  raw 

Velvets,  cotton,  unbleached  silk  .... 
Wines,  still,  in  casks  containing  more  than 

14  %  alcohol 

Woolen  or  worsted  clothing 


¥4^  per  lb. 

40%  ad  valorem 

22 J?  per  sq.  yd.  &  40%  ad  valorem 

18^  per  sq.  yd.  &  40%  ad  valorem 

28^  per  sq.  yd.  &  40%  ad  valorem 

60^  per  sq.  yd.  &  40%  ad  valorem 

$4.50  per  lb.  &  20%  ad  valorem 

30%  ad  valorem 

9^  per  sq.  yd.  &  25%  ad  valorem 

15^  per  piece  &  35%  ad  valorem 

$1.75  per  doz.  &  20%  ad  valorem 

50%  ad  valorem 

%^  per  lb. 

10  j?  per  sq.  ft. 

$4  per  doz.  pair 

2^2^  per  lb. 

$4  per  ton 

50%  ad  valorem 

45^  per  lb.  and  60%  ad  valorem 

60%  ad  valorem 

Free 

9^  per  lb.  &  25%  ad  valorem 

60^  per  gal. 

44^  per  lb.  &  60%  ad  valorem 


190 


ELEMENTS  OF  BUSINESS  ARITHMETIC 


PROBLEMS 

1.  What  is  the  duty  on  60  yd.  of  silk,  weighing  4  oz.  to  the  yard, 
invoiced  at  $  1.50? 

2.  What  is  the  duty  on  a  consignment  of  26  doz.  men's  kid  gloves, 
invoiced  at  $22.50  per  dozen  ? 

3.  If  silk  laces  are  invoiced  at  7540  francs,  what  will  be  the  duty? 

4.  What  will  be  the  duty  on  an  importation  of  |646  worth  of  skins? 

5.  What  is  the  duty  on  an  invoice  of  642  lb.  glue;  240  lb.  alcohol; 
20  doz.  brushes,  at  25)^  each  ;  and  450  lb.  extract  of  dye  wood? 

6.  What  is  the  duty  on  14  doz.  sword  blades,  worth  $7.40  per  dozen, 
and  16  doz.  razors  worth  |4.50  per  dozen? 

7.  For  what  must  I  sell  hose,  if  they  are  invoiced  at  $2  per  dozen, 
in  order  to  gain  20%  after  paying  the  duty? 

8.  A  merchant  imported  1420  yd.  of  plush  (f  yd.  wide),  invoiced 
at  2  francs  a  square  yard.  At  what  price  per  yard  must  he  sell  it  to  gain 
25%,  after  paying  $65  freight  charges  and  the  import  duty? 

9.  What  is  the  duty  on  650  yd.  Axminster  carpet,  30  in.  wide,  if  the 
invoice  price  is  40^  per  yard?    For  what  must  it  be  sold  to  gain  33^ 7o? 

10.  1000  boxes  (100  in  each)  of  cigars  weighing  1200  lb.  net  cost 
$13,675,  delivered.  If  the  freight  and  charges,  other  than  the  duty, 
amount  to  $45,  what  was  the  invoice  price  of  the  cigars? 

11.  After  allowing  10%  for  leakage,  a  wine  merchant  paid  $1768  duty 
on  24  casks  of  wine  at  $2  per  gallon.  How  many  gallons  did  the  casks 
originally  contain? 

12.  What  will  be  the  duty  on  250  boxes  plate  glass,  each  containing 
25  plates,  16  x  20  in.? 

13.  Find  the  amount  of  the  duty  and  the  total  cost  of  the  following: 

Manchester,  England,  Jan.  14,  19 — . 
Messrs.  A.  T.  BOYD  &  CO., 

New  York,  N.Y. 

Bought  of  WILLIAM  &  FIRTH 


Terms 

:  30  da. 

^ 

565  yd.  Silk  Velvet       1  12s. 
3050  yd.  Linen                     45.   Sd. 
1450  yd.  Cotton  Webbing     Is. 
4250  yd.  Wilton  Carpet       125.  6c?. 

TAXES  AND  DUTIES 


191 


14. 

Manifest  No.  675  Invoiced  at  Bradford,  Eng.,  June  12,  19 — . 

INWARD  FORWARD  ENTRY  OF  MERCHANDISE 

Imported  by  Gates  &  Warman.  In  the  Steamer  Columbia. 

Amos  Peterman,  Master.  From  Liverpool,  Eng. 

Arrived  July  15, 19 — . 


Marks 

No. 

Packages  and 
Contents 

Quantity 

Free 
List 

Valub 

Ad 

Valorem 

Duty 

Specific 
Duty 

TOTAI. 

Duty 

A.A. 

1356 
756 

4  cases  Woolen 

Clothing 
6  cases  Cotton 

Hose 

2400  lb. 
192  doz. 

£342.12.0 
£74.10.0 

? 
? 

? 
? 

? 
? 

$ 

•$ 

$ 

$ 

XVII 

INTEREST 

160.  Loaning  Money.  When  money  is  loaned,  a  charge 
is  made  for  its  use.  This  charge  or  rental  for  the  use  of 
money  is  called  interest.  Interest  is  always  a  certain  per 
cent  of  the  amount  loaned,  and  the  rate  is  the  rental  for  a 
period  of  one  year. 

Thus,  15000  loaned  at  6%  would  mean  that  6%  of  the 
$5000,  or  $300,  must  be  paid  for  its  use  each  year.  Of 
course,  one  half  of  that  amount,  or  $  150,  would  be  the  in- 
terest for  six  months,  and  twice  that  amount,  or  $  600,  would 
be  the  interest  for  two  years,  etc. 

161.  Notes  and  Mortgages.  When  loans  are  made,  a 
written  instrument  or  contract  called  a  promissory  note  is 
signed  by  the  persoij  receiving  the  money.  It  is  a  written 
promise  to  pay  the  amount  named  at  a  certain  time,  usually 
with  interest  from  date  of  the  note,  or  from  the  date  agreed 
upon  for  payment,  called  maturity^  and  contains  an  acknowl- 
edgment of  the  receipt  of  value  therefor. 

The  payment  of  a  note  is  often  made  more  secure  by  the 
signature  of  other  persons  as  guarantors  and  known  as  in- 
dorsers,  or  by  conveyance  of  property,  real  or  personal,  con- 
ditioned upon  the  nonpayment  of  the  note.  When  property 
is  thus  pledged,  the  instrument  of  conveyance  is  called  a 
mortgage.  If  for  personal  property,  it  is  a  chattel  mortgage  ; 
if  for  real  estate,  a  real  estate  mortgage  deed.  In  either  case, 
the  title  only  to  the  property  is  transferred,  possession  to  be 
given  upon  nonpayment  of  the  note,  and  upon  the  success 

192 


INTEREST  193 

of  legal  proceedings  known  as  foreclosure.  The  one  who 
signs  the  note  is  called  the  maher^  and  the  one  to  whom  the 
note  is  to  be  paid  is  called  the  payee. 


/i  lA^-^^  C4''ri^..^.yL.d^t7-ny,  '7(ayn..4'-eLJ .  ///oyy  f,  /f 


.^jz^-^ZiiCi-^  /Z'gl^k.-tr-yz^itL^  ~Aj^-tZ^^yi,^k^,^v-f'  Cyt^yi.-^e6£yt<d^trny^ 


/!/^a,■€U<..^^^4.C'C^^c^^ 


-<^'i<2^- 


A  Promissory  Note. 

162.  Bonds.  For  loans  made  to  corporations,  the  note  is 
called  a  hond.  These  may  be  either  governmental,  as  city 
or  county,  or  private,  often  called  industrial  bonds.  In 
notes  or  bonds  extending  through  a  period  of  years,  the  in- 
terest is  often  provided  for  in  separate  notes  for  each  interest 
payment,  which  are  attached  to  the  principal  note,  and  are 
known  as  interest  coupons.  These  bonds  are  usually  issued 
for  the  purpose  of  raising  money  for  improvements,  develop- 
ment of  property,  etc.  They  pledge  the  property  of  the  corpo- 
ration as  security,  and  are  generally  sold  in  the  open  market. 

163.  Compound  Interest.  In  Compound  Interest^  unpaid 
interest  is  added  to  the  principal  at  the  end  of  each  interest 
period.  The  whole  amount  is  then  considered  as  a  new  prin- 
cipal^ upon  all  of  which  interest  thereafter  is  to  be  paid. 
The  interest  and  principal  are  thus  compounded  or  united. 
Thus,  1100  at  6^  would  amount  to  $106  at  the  end  of  the 
first  year,  to  1112.36  at  the  end  of  the  second  year,  to  $119.10 
at  the  end  of  the  third  year,  etc.  Money  loaned  at  compound 
interest  increases  so  rapidly  as  to  more  than  double  itself  at 
6  %  in  12  years. 


194  ELEMENTS  OF  BUSINESS  ARITHMETIC 

164.  Periodic  Interest.  When  interest  due  at  the  close  of 
each  interest  period  begins  to  draw  simple  interest  until  the 
obligation  is  paid,  it  is  said  to  be  periodic  interest.  That  is, 
the  interest  on  the  principal  for  each  time  period  draws  in- 
terest ;  but  this  interest  on  the  interest  does  not  become  due 
until  the  note  is  paid,  and  therefore  does  not  itself  draw  in- 
terest ;  i.e.  it  is  not  compounded.  Thus,  a  |100  note,  if  paid 
at  the  end  of  the  first  year,  would  amount  to  $106,  at  end  of 
second  year  to  $112.36,  third  year  to  $119.08,  etc. 

If  the  interest  period  is  one  year,  it  is  annual;  if  six  months, 
semiannual ;  and  if  three  months,  quarterly  interest. 

$100  at  6%  simple  interest  would  amount  in  ten  years  to 
$160;  at  annual  periodic  interest  to  $176.20;  and  at  com- 
pound interest  to  $179.08. 

165.  Legal  Limitations  to  Interest.  When  the  rate  of  in- 
terest is  not  stipulated  in  the  note  or  contract,  the  rate  fixed 
by  law  is  collectable.  Such  rate  is  known  as  the  legal  rate., 
and  varies  in  different  states  and  territories. 

Most  states  establish  a  limit  to  the  rate  which  may  be 
charged  even  when  agreed  to  by  both  parties.  This  is  known 
as  the  contract  rate.  Interest  charged  in  excess  of  the  amount 
allowed  by  law  is  usury.  While  several  states  have  no  pro- 
visions regarding  usury,  most  states  forbid  it.  Various  pen- 
alties are  prescribed,  such  as  the  forfeiture  of  principal  and 
interest,  of  principal,  of  interest,  or  of  excess  of  interest  over 
the  legal  rate. 

The  collection  of  compound  interest  cannot  usually  be  en- 
forced by  legal  process,  but  its  payment  or  receipt  is  not 
illegal. 

In  some  states  periodic  interest  may  be  collected  when  in 
the  contract,  while  in  others  it  cannot  be  enforced.  When 
each  interest  payment  is  put  in  the  form  of  a  note,  as  a  coupon, 
it  may  be  collected  in  any  state  and  is  not  considered  usury. 


INTEREST  195 

In  order  to  prevent  undue  hardship,  most  states  have,  in 
the  past,  allowed  a  creditor  three  days  after  the  stipulated 
date  of  maturity,  before  a  note  might  be  collected  through 
legal  process.  These  are  known  as  days  of  grace.  The  cus- 
tom, however,  is  dying  out,  and  most  states  have  repealed 
the  law.  Improved  communication  leaves  less  necessity  for 
days  of  grace  than  formerly. 

166.  Common  and  Exact  Interest.  In  computing  the  time 
of  notes  or  other  interest-bearing  obligations,  the  method  of 
compound  subtraction  (97)  is  used.  It  is  the  common  prac- 
tice to  count  a  month  as  uniformly  30  days  and  the  year 
360  days.  If  the  note  reads  "  days  after  date,"  the  date  of 
maturity  is  usually  computed  to  the  exact  day  ;  if  "  months 
after  date,"  then  the  same  day  of  the  month  in  the  appropri- 
ate month. 

The  method  still  in  use  by  the  United  States  government 
and  by  the  more  conservative  business  men  takes  account  of 
the  exact  number  of  days  in  the  intervening  calendar  months, 
and  the  days  are  365ths  of  a  year.  This  is  known  as  exact 
interest.  Exact  time  between  given  dates  and  exact  interest 
at  any  rate  on  various  amounts  for  any  number  of  dates,  are 
usually  found  by  reference  to  "  Exact  Interest  Tables,"  with 
computations  already  made. 

Interest  found  by  the  60-day  method,  or  by  any  method 
using  30  days  as  a  month,  counts  a  year  as  360  days.  It  is, 
therefore,  ^l-g-,  or  y^^,  greater  than  "  exact "  interest  for  the 
same  period.  Adding  ^^  of  itself  to  "  exact "  interest  would 
likewise  equal  "  30-day  "  interest. 

167.  Finding  Interest.  When  the  period  of  a  note  or  other 
obligation  is  an  exact  number  of  years,  the  problem  is  one  of 
simple  percentage.  If  the  interest  rate  is  6  %  and  the  time 
three  years,  18  %  of  the  amount  loaned  would  be  the  inter- 
est ;  if  8  %  for  4  years,  32  %  of  the  amount  loaned,  etc. 


196  ELEMENTS  OF  BUSINESS  ARITHMETIC 

But  most  interest  computations  in  business  have  to  do  with 
short-time  loans  and  discounts.  Hence  they  involve  chiefly 
months  and  days.  On  account  of  the  frequent  occurrence 
of  6  %  as  a  rate  of  interest  in  business,  and  because  of  the 
simple  relation  6  bears  to  12  (the  number  of  months  in  the 
year)  and  the  easy  fractional  parts  that  relation  makes  in 
computing  time,  interest  on  any  sum  for  a  short  period  is  most 
easily  and  accurately  calculated  as  though  at  6  9^,  and  then 
changed  to  the  required  rate. 

168.  Six  Per  Cent  Interest  for  60  Days.  If  every  12  months 
6  %  of  the  amount  of  the  loan  must  be  paid  as  interest,  every 
two  months  -f^  (e)  ^^  ^^^^'  or  1  %,  must  be  paid.  Then  1  % 
of  the  amount  loaned  would  be  the  interest  for  2  months,  or 
60  days. 

Thus,  1840  at  6%  for  60  days  is  18.40;  and  11726  at  6% 
for  60  days  is  f  17.26.  From  this  it  will  be  seen  that  the 
interest  on  any  sum  at  %%  for  60  days  is  as  many  cents  as 
dollars  loaned ;  the  same  is  true  of  4  %  interest  for  three 
months  and  3%  interest  for  four  months.  These  rates  are 
usable  in  savings  banks  and  for  interest  on  government 
bonds. 

169.  Six  Per  Cent  for  Any  Time. 

Problem.  —  Find  the  interest  on  $840  for  1  yr.  5  mo.  6  da. 

$8.40,  interest  for  2  mo. 
$67.20  ($8.40  X  8),  interest  for  16  mo. 
4.20  (^  of  $8.40),  interest  for  1  mo. 

^  (-ji^  of  $8.40),  interest  for  6  da. 

$72,24,  interest  for  1  yr.  5  mo.  6  da. 

To  find  the  interest  for  years  and  months,  first  reduce  to 
months.  Then  take  such  a  simple  multiple  of  the  interest 
for  two  months  as  will  make  the  required  time,  if  for  even 
months.     If  for  odd  months,  add  \  of  the  two  months'  interest 


J?,  r^ 


INTEREST  197 

to  the  interest  for  one  less  than  the  required  number  of 
months. 

For  days,  fractional  parts  of  the  interest  for  2  months 
should  be  taken,  using  simple  fractional  parts  wherever  prac- 
ticable and  doing  as  much  mentally  as  possible.  Thus,  the 
interest  for  15  days  is  ^  that  for  60  days;  for  three  days  J  of 
■f-Q  of  it ;  for  5  days  -^^  of  it ;  for  7  days  -^q  of  it  plus  J  of  the 
latter  (6  days  plus  1  day),  etc. 

170.  Interest  at  Any  Rate.  If  any  other  rate  had  been 
given  in  the  problem  in  the  above  section,  the  process  would 
be  identical  to  the  point  of  finding  the  interest  at  6  %  for 
the  given  time.  If  the  rate  were  5%,  the  required  interest 
would  be  1  less  than  $72.24,  or  f  60.20 ;  if  7%,  884.28,  or  1 
more;  if  4%,  $48.16,  or  1  less;  if  8%,  $96.82,  or  1  more; 
if  4J  %,  $54.18,  or  {  less,  etc. 

Hence  to  find  interest  at  any  rate  :  Find  the  total  interest 
at  Q%  and  increase  or  decrease  it  hy  the  fractional  'part  that 
the  required  rate  is  greater  or  less  than  6  %. 

171.  To  find  the  Face  of  a  Note. 

Problem.  —  The  interest  on  a  note  for  8  mo.  at  5  %  was 
$40.     Find  the  face  of  the  note. 

$40,  interest  at  5  %  for  8  mo. 

8,  interest  at  1  %  for  8  mo.  (|  of  $40). 

$48,  interest  at  6  %  for  8  mo. 

$12,  interest  at  6%  for  2  mo.  (\  of  $48). 

Then,  $1200  ($12  x  100)  is  the  face  of  the  note. 

When  the  interest  is  given  at  any  rate,  first  change  it  to 
what  it  would  amount  to  at  Q%.  This  being  the  interest  at 
6  %  for  the  given  time,  such  a  fraction  of  it  should  he  taken  as 
will  give  the  interest  for  two  months  only.  Since  this  would 
represent  as  many  cents  as  dollars  loaned,  100  times  that 
amount  would  he  the  face. 


198  ELEMENTS  OF  BUSINESS  ARITHMETIC 

172.  To  find  Note  Period. 

Pkoblem.  —  The   interest  on  |680  at  4|  %  was  120.40. 
How  long  did  the  note  run  ? 
120.40,  interest  at  4J%. 

6.80,  interest  at  li  %  Q  of  120.40). 
127.20,  interest  at  6%. 

6.80,  interest  on  S680  for  2  mo. 

3.40,  interest  on  $680  for  1  mo. 
f  2T.20  -h  13.40  =  8,  the  length  of  note  period  in  months. 

The  interest  given  is  changed  to  the  interest  at  6fo-  The 
interest  on  the  given  principal  for  1  month  atQ^fo  is  then  found, 
and  divided  into  the  interest  for  the  whole  term  at  Qf/o-  The 
quotient  is  the  number  of  months  the  note  runs. 

173.  To  find  the  Rate  of  Interest. 

Problem.— If  a  note  for  1660  draws  138.50  in  1  yr.  2 
mo.,  what  is  the  rate  of  interest? 

1660  for  1  yr.  2  mo.  would  earn  146.20  at  6  %,  and  l  of 
that,  or  $7.70  at  1%.  The  required  rate  is  as  many  per 
cent  as  $7.70  is  contained  times  in  $38.50,  or  5  %. 

The  interest  at  l^o  for  the  given  time,  divided  into  the  given 
interest,  will  give  the  rate  required. 

174.  To  find  what  Sum  will  produce  a  Given  Amount. 
Problem.  —  What  will  produce  $3955.95  at  4  %  for  1  yr. 

6  mo.  18  da.  ? 

$.01,      interest  $1  will  produce  in  2  mo.  at  6%. 
.09,      interest  on  $1  for  1  yr.  6  mo.  (18  mo.). 
.0025,  interest  on  $1  for  15  da. 
.0005,  interest  on  $1  for  3  da. 
.093,    interest  on  $1  for  1  yr.  6  mo.  18  da. 
.031,    interest  on  $1  at  2  %  (i  of  $.093). 
.062,    interest  on  $1  for  1  yr.  6  mo.  18  da.  at  4  %. 

$1  will  amount  to  $1,062  in  1  yr.  6  mo.  18  da.  at  4%. 


J 


INTEREST  199 

13955.95  --11.062  =  3725,  or  the  number  of  dollars  which 
will  amount  to  $3955.95  at  4%  for  the  given  time. 

The  amount  of  ^1  for  the  given  time  and  rate  is  first  found. 
The  number  of  times  that  is  contained  in  the  given  amount  is 
the  number  of  dollars  required  to  produce  that  amount. 

175.  Other  Methods.  Interest  tables,  with  the  interest 
already  calculated  for  any  number  of  dollars  for  any  number 
of  days  at  given  rates,  are  largely  used  by  banks,  trust  com- 
panies, insurance  offices,  etc.  The  calculated  interest  is 
arranged  in  tabulated  form  for  convenient  reference,  and 
the  work  of  computing  interest  is  thereby  greatly  lessened. 
Many  different  arrangements  of  such  tables  are  published. 

The  formal  6  %  method  is  often  used  by  accountants,  par- 
ticularly when  the  time  is  more  than  1  year.  It  is  based  on 
finding  the  interest  on  $1  for  the  given  time,  and  multiply- 
ing that  by  the  number  representing  the  dollars  loaned. 
Thus,  $1  could  bear  $.06  interest  in  one  year,  $.005  interest 
in  one  month,  1.000 J  interest  for  one  day,  *etc. 

The  interest  at  any  other  rate  would  then  be  found  in  the 
same  way  as  by  the  60-day  method.     (Sec.  169.) 

176.  Partial  Payments.  Payments  on  long-term  notes, 
made  from  time  to  time,  are  known  as  partial  payments. 
These  payments,  with  their  dates,  are  usually  indorsed  on 
the  back  of  the  note. 

By  what  is  known  as  the  United  States  Rule^  each  payment 
is  deducted  from  the  amount  of  the  debt  at  the  time  of  the 
payment,  and  the  remainder  or  mqw  principal  will  bear  simple 
interest  until  the  next  payment.  If  a  payment  is  not  as  large 
as  the  accrued  interest,  it  cannot  reduce  the  debt.  It  is 
therefore  disregarded,  and  added  to  the  next  payment. 

By  the  Merchants'  Rule^  the  sum  of  all  the  partial  pay- 
ments, together  with  the  interest  on  each  of  them  from  its 
date  until  maturity  or  date  of  settlement  of  the  note,  is  sub- 


200  ELEMENTS  OF  BUSINESS  ARITHMETIC 

tracted  from  the  sum  the  original  principal  would  amount 
to,  at  simple  interest,  on  the  date  of  settlement. 

Because  merchants  and  bankers  prefer  short-term  notes, 
with  renewal  (new  note)  for  part  of  debt  remaining  unpaid, 
long-term  notes,  except  when  in  the  form  of  mortgage  notes 
or  bonds,  are  disappearing  from  business.  Partial  payments 
on  mortgage  notes  and  bonds  may  sometimes  be  made,  but 
only  in  stipulated  amounts  and  at  specified  dates,  while  the 
interest  is  covered  by  coupon  notes,  which  are  treated  as 
separate  obligations.  On  account  of  its  occasional  use  in 
some  sections  of  the  country,  the  following  illustration  of 
partial  payments  is  given  : 

Problem.  —  Omaha,   Neb.,  Jan.  1,  1920. 

$500.00 

On  demand,  I  promise  to  pay'  C.   L.  Webster,  or  order. 

Five  Hundred  and  00/100  Dollars,  value  received,  with 
interest  from  date  at  6  %.  J.  L.  Fernby. 

If  endorsed  March  1, 1920,  $105;  July  1,  1920,  |108;  Jan. 
1,  1922,  120.     What  is  due  at  settlement,  Nov.  1,  1922? 
1500,     original  debt. 
5^     interest  2  mo.  (Jan.  1  to  March  1,  1920). 

505,     due  March  1,  1920. 

105,     payment  Jan.  1,  1920. 

400,     unpaid  debt  or  new  principal. 
8,     interest  4  mo.  (March  1  to  July  1,  1920). 


408,     due  July  1,  1920. 
108,     payment  July  1,  1920. 
300,     unpaid  debt  July  1,  1920. 

$27,  interest  for  18  mo.  (July  1, 1920  to  Jan.  1, 1922), 
120,  payment  Jan.  1,  1922,  deferred. 
42,     interest  for  28  mo.  (July  1,  1920  to  Nov.  1, 1922). 
842,     due  Nov.  1,  1922. 
20,     payment  Jan.  1,  1922. 
$  322,     net  amount  due  at  settlement. 


INTEREST  201 


PROBLEMS 


1.  Mr.  Williams  has  the  use  of  $  240  for  1  yr.  at  6  %.     What  amount 
of  interest  does  he  pay  ? 

2.  A  merchant  borrows  $260  for  1  yr.  at  5%.     How  much  does  he 
pay  for  its  use  ? 

3.  At  4  %  what  must  I  pay  for  the  use  of  $  600  for  1  yr.  ? 

4.  Mr.  Constable  borrowed  $750  for  1  yr.  at  8%.     How  much  inter- 
est did  he  pay  ? 

5.  A  note  of  $  324  for  2  mo.  at  6  %  will  draw  how  much  interest  ? 

6.  A  banker  loans  $  240  for  4  mo.  at  6  %.     How  much  interest  does 
he  receive  ? 

7.  A  farmer  sold  a  horse,  taking  a  note  of  $  175  for  6  mo.  at  6  %  in 
payment.     Find  amount  due  at  the  end  of  the  time. 

8.  A  manufacturer  borrows  $  1250  for  3  mo.  at  6  %.     How  much 
interest  does  he  pay  ? 

9.  B  borrows  $  75  for  1  mo.  at  6  %.     What  amount  is  due  at  the  end 
of  the  time  ? 

Find  Interest: 

10.  On  $  150  for  1  mo.  @  6  %. 

11.  On  1720  for  4  mo.  @  6%. 

12.  On  1480  for  3  mo.  @  6%. 

13.  On  170  for  5  mo.  @  6  %. 

14.  On  $65  for  10  da.  @  6  %. 

15.  On  $120  for  45  da.  @  6  %. 

16.  On  $160  for  6  da.  @  6%. 

17.  On  $  142  for  3  da.  @  6  % 

18.  On  $85  for  6  mo.  @  8  %. 

19.  On  $460  for  8  mo.  @  5  %. 

20.  On  $  540  for  10  mo.  @  7  %. 

21.  On  $60  for  2  yr.  6  mo.  @  6%. 

22.  On  $54  for  1  yr.  2  mo.  15  da.  @  6%. 

23.  On  $480  for  75  da.  (60  +  15)  @  6  %. 

24.  On  $76  for  63  da.  (60  +  3)  @  6  %. 

25.  On  $960  for  90  da.  (60  +  30)  @  6  %. 


202  ELEMENTS  OF  BUSINESS  ARITHMETIC 

26.  On  124  for  85  da.  (60  +  10  +  15)  @  6  %. 

27.  On  $36  for  123  da.  (2  60's  +  3)  @  6%. 

28.  On  $48  for  54  da.  (60  -  6)  @  6  %. 

29.  On  $54  for  36  da.  (30  +  6)  @  6  %. 

30.  On  $64  for  42  da.  (30  +  12)  @  6%. 

31.  On  $480  for  18  da.  (3  6's)  @  6%. 

32.  On  $730  for  72  da.  (60  +  12)  @  8%. 

33.  On  $75.60  for  87  da.  @  8%. 

34.  On  $  1244  for  8  mo.  20  da.  @  7  %. 

35.  On  $450  for  1  mo.  20  da.  @  5 %;  4 %. 

36.  On  $860  for  3  mo.  15  da.  @  4%;  8  %. 

37.  The  interest  on  a  note  was  $40,  the  rate  6%,  and  the  time  8  mo. 
Find  the  face. 

38.  What  was  the  face  of  a  note,  if  the  time  was  5  mo.,  the  interest 
$16,  and  the  rate  6  %? 

39.  The  time  was  7  mo.,  the  rate  6  %,  and  the  interest  $  13.30.     What 
was  the  face  of  the  note  ? 

40.  I  paid  a  bank  $3.44  interest.     I  had  the  money  8  mo.  at  6  7o' 
How  much  did  I  have  ? 

41.  The  interest  at  6%  on  a  note  for  1  yr.  3  mo.  was  $  3.15.     What 
was  the  face  ? 

42.  What  was  the  face  of  a  note  given  for  5  mo.  at  6  %,  if  the  interest 
was  $19.50? 

43.  The  interest  on  $  680  at  6  %  was  $  27.20.     How  long  did  the  note 
run? 

44.  How  long  did  a  note  for  $  800  at  6  %  run,  if  the  interest  was  $38? 

45.  Mr.  Wood  paid  $20.90  interest  at  6  %  on  a  note  of  $380.     How 
long  did  he  have  it  ? 

46.  If  the  principal  was  $  120,  rate  6  %,  and  interest  $  7.80,  what  was 
the  time  ? 

47.  A  note  of  $  150  drew  $  8  interest  at  8  %.     How  long  had  it  run  ? 

48.  The  interest  on  a  note  of  $  180  at  8  %  was  $  12.    How  long  had  it 
run? 

49.  If   a  note  for  $650  draws  an  interest  of  $39  in  1  yr.,  what  is 
the  rate? 


INTEREST  203 

50.  If  I  pay  1 60  on  $  1000  for  1  yr.,  what  is  the  rate  ? 

51.  At  what  rate  would  $  710  produce  $  17.75  in  5  mo.  ? 

52.  The  interest  on  $  640  for  4  yr.  was  $  179.20.     What  was  the  rate? 

53.  What  rate  was  charged  if  a  merchant  paid  $  43|  for  the  use  of 
$  650  for  16  mo.  ? 

54.  A  farmer  deposited  $  425  in  the  bank  and  at  the  end  of  8  mo. 
drew  out  $  437.75.     What  rate  of  interest  did  he  receive  ? 

55.  The  amount  due  on  a  note  of  $  120  at  the  end  of  8  mo.  was  f  124. 
What  rate  of  interest  was  charged? 

56.  Principal  $200,  time  3  yr.,  interest  $  30.     Rate? 

57.  Principal  $  180,  rate  4  %,  interest  $  14.40.     Time  ? 

58.  Time  8  mo.,  rate  6  %,  interest  1 7.     Principal? 

59.  Principal  1 76,  time  3  yr.,  amount  $ 87.40.     Rate? 

60.  Principal  $  138.40,  rate  4%,  time  2  yr.  6  mo.  15  da.     Interest? 

61.  Amount  $3961.26,  rate  4  %,  time  1  yr.  6  mo.  18  da.     Principal? 

62.  Amount  $  502.09,  rate  6  %,  principal  $  460.     Time  ? 

63.  Time  4  yr.  6  mo.,  rate  5  %,  amount  1 7181.80.     Interest? 

64.  Principal  $7548,  time  3  mo.  5  da.,  interest  $119.51.     Rate? 

65.  Principal  $  900,  rate  5  %,  interest  $  56.25.     Time  ? 

66.  Principal  $ 72,  rate  6 %,  time  2  yr.  6  mo.     Interest? 

67.  Principal  $460,  rate  4  %,  time  1  yr.  5  mo.  6  da.     Interest? 

68.  Principal  $  126,  rate  7|  %,  time  2  yr.  3  mo.  18  da.     Amount? 

69.  Principal  $  640,  rate  6  %,  interest  $  86.40.     Time  ? 

70.  Rate  6%,  time  1  yr.  6  mo.  24  da.,  interest  $ 338.40.     Principal? 

71.  Principal  $  3200,  time  2  yr.  6  mo.  6  da.,  amount  $  3602.67.     Rate  ? 

72.  Principal  $324,  rate  5  %,  time  1  yr.  4  mo.  15  da.     Interest? 

73.  Principal  $ 960,  rate  5%,  interest  $114.     Time? 

74.  Principal  $  370,  time  1  yr.  6  mo.  15  da.,  interest  $  34.23.     Rate  ? 

75.  Rate  8%,  time  2  yr.  8  mo.  24  da.,  interest  $125.73.     Principal? 

76.  Principal  $2125,  time  11  mo.  24  da.,  amount  $2250.38.     Rate? 

77.  Principal  $  1440,  rate  6  %,  amount  $  2100.     Time  ? 

78.  A  note  for  $450,  dated  April  15,  1914,  payable  on  demand,  with 
interest  at  6  %,  bears  the  following  indorsements  :  June  1,  $  50  ;  Oct.  10, 
$  100 ;  Nov.  15,  $  35  ;  Dec.  10,  $  75.     What  is  due  Feb.  12,  1915  ? 


^Ni 


204  ELEMENTS  OF  BUSINESS  ARITHMETIC 

79.  What  is  due  May  15,  1916,  on  a  note  for  $  600,  dated  July  12, 
1914,  bearing  5  %  interest,  with  the  following  indorsements :  Sept.  21, 
1914,  $75;  Feb.  18,  1915,  |200;  May  1,  1916,  $65? 

80.  A  note  for  $  1500,  dated  Nov.  14,  1915,  and  due  Nov.  14,  1917, 
with  interest,  has  the  following  indorsements:  Jan.  27,  1916,  $340; 
May  13,  1916,  $  250  ;  Sept.  16,  1916,  $25  ;  Feb.  17,  1917,  $20.  What  is 
due  at  maturity  ? 

Compute  exact  interest  on  the  following.  (See  Sec.  166  and  Prob.  1, 
at  close  of  Chapter  IX.) 

81.  $  720  for  73  da.  at  8  %. 

82.  $460  for  63  da.  at  5%. 

83.  $860  for  1  mo.  20  da.  at  8  %. 

84.  $800  from  Jan.  14,  1900  to  June  18,  1900,  at  8  %. 

85.  $  760  from  May  16  to  July  27,  at  7  %. 

86.  $  154.80  from  Jan.  15  to  Feb.  16,  at  5%. 

87.  $  172.50  from  Nov.  12,  1891  to  July  6,  1892,  at  4%. 

88.  $465.20  from  Jan.  29,  1901  to  Jan.  1,  1903,  at  5  %. 

89.  $900  from  June  15,  1901  to  Jan.  15,  1904,  at  4|%. 

90.  $  140  from  May  1  to  Sept.  25,  at  8  %.   • 

91.  $  560  from  Jan.  24  to  Dec.  16,  at  5  %. 

92.  $37.50  from  April  3  to  May  1,  at  8%. 

93.  $  184.50  from  June  4,  1901  to  July  16,  1903,  at  7%. 


XVIII 

BANKING  AND  DISCOUNT 

177.  Banking.  A  hank  is  a  business  institution  for 
the  receiving  and  safe-keeping  of  money,  and  the  making 
of  loans.  It  also  deals  in  credits,  discounts  negotiable 
paper,  collects  accounts,  and  makes  payments  in  other 
cities. 

A  bank  is  usually  a  corporation,  chartered  by  law,  and  is 
subject  to  supervision  by  the  federal  or  state  authorities,  as 
a  means  of  safeguarding  the  interests  of  depositors. 

178.  National  Banks.  Banks  organized  under  "  The  Na- 
tional Banking  Act "  are  entitled  to  the  use  of  the  word 
"National"  in  their  name.  Besides  transacting  a  general 
banking  business,  they  may  issue  "bank  notes,"  which  are 
secured  by  the  deposit  of  United  States  Government  bonds 
with  the  United  States  Treasurer.  The  government  imposes 
certain  limitations  as  to  the  modes  of  doing  business,  the 
amount  of  the  reserve,  etc.  It  requires  uniform  reports, 
makes  regular  examinations  into  the  bank's  business,  and 
prescribes  penalties  for  violation  of  the  requirements  of  the 
bank  law. 

179.  State  and  Private  Banks.  State  banks  are  organized 
under  the  provisions  of  the  statutes  of  the  various  states, 
and  are  subject  to  such  restrictions  as  those  statutes  impose. 
State  banks  do  not  issue  circulating  notes  and  are  not  re- 
quired, therefore,  to  hold  United  States  bonds.  They  are 
not  usually  so  carefully  and  frequently  examined,  are  not 

205 


206  ELEMENTS  OF  BUSINESS  ARITHMETIC 

required  to  make  such  rigid  reports,  and  are  not  restricted 
in  so  many  ways  in  the  conduct  of  their  business,  as  are  the 
national  banks. 

Individuals  or  firms  may  operate  private  banks  and  trans- 
act a  general  banking  business,  i.e.  receive  deposits,  make 
loans,  deal  in  exchange  and  other  commercial  paper,  etc. 
They  are  usually  subject  to  statutory  provisions  designed  to 
insure  the  safety  of  funds  deposited. 

180.  Savings  Banks.  Savings  banks  are  also  organized 
under  state  laws.  Receiving  small  as  well  as  large  deposits, 
and  paying  interest  thereon,  they  are  designed  to  promote 
economy  and  encourage  thrift. 

The  money  deposited  begins  to  draw  interest  on  the  first 
day  of  the  following  month  or  of  the  following  quarter, 
according  to  the  rules  of  the  particular  bank.  The  days 
on  which  deposits  begin  to  earn  interest  are  known  as  interest 
days.  When  interest  days  are  quarterly,  they  are  January  1, 
April  1,  July  1,  and  October  1. 

Interest  is  usually  allowed  only  on  those  sums  which  have 
been  on  deposit  for  the  full  time  between  interest  days. 
Thus,  the  lowest  daily  balance  in  the  month  or  quarter,  not 
counting  fractional  parts  of  a  dollar,  is  the  amount  upon 
which  interest  is  computed.  The  interest  is  usually  declared 
semiannually.  If  not  paid,  it  is  credited  to  the  depositor's 
account  and  draws  interest  thereafter. 

181.  Savings  Bank  Accounts. 

Problem  1.  —  The  dividends  of  interest  at  a  savings 
bank  are  declared  semiannually.  A  customer  deposited 
June  5,  1300;  Aug.  10,  l|150;  Oct.  1,  $100;  Dec.  10,  $20. 
No  withdrawals  having  been  made,  what  was  due  Jan.  1, 
following,  if  interest  be  reckoned  on  the  deposits  from  the 
first  of  each  quarter  at  4  %  ? 


BANKING  AND  DISCOUNT 


207 


Dates 

Deposits 

Daily 

Balances 

Interest 
Days 

Smallest 
Quarterly 
Balances 

Quarterly 
Interest 

June     5 
Aug.  10 
Oct.      1 
Dec.    10 
Jan.      1 

$300 

150 

100 

20 

$300 
450 
550 
570 
570 

July  1 
Oct.   1 
Jan.  1 

$300 

550 

$3.00 

5.50 

8.50 

570.00 

$578.50 

Solution.  —  Since  interest  begins  on  the  first  of  each 
quarter,  the  deposit  of  Aug.  10  will  not  begin  to  draw 
interest  until  the  beginning  of  the  second  quarter.  The 
interest  on  $300  for  one  quarter  is  $3.  The  deposits  of 
Aug.  10  and  Oct.  1,  together  with  the  balance  for  the  first 
quarter  ($550),  will  draw  interest  for  the  second  quarter. 
The  deposit  of  Dec.  10  will  not  draw  interest  until  the 
beginning  of  the  third  quarter. 

If  interest  days  are  monthly,  the  above  account  would 
balance  as  follows : 


Dates 

Deposits 

Interest 
Days 

Smallest  Monthly 
Balances 

Monthly 
Interest 

June    5 
Aug.  10 
Oct.      1 
Dec.   10 

$300 

150 

100 

20 

July   1 
Aug.  1 
Sept.  1 
Oct.    1 
Nov.  1 
Dec.    1 

$300 
300 

450 

550 
550 

$1.50 
1.50 
2.25 
2.75 
2.75 

Jan.    1 

570 

2.75 

13.50 

4.50 

9.00 

570.00 

$579.00 

208 


ELEMENTS  OF  BUSINESS  ARITHMETIC 


The  aggregate  interest  on  the  smallest  monthly  balances  at 
6  %  is  found  to  be  $13.50.  At  4  %  it  is  §9.  $570  +  $9  = 
$579,  the  balance  due  Jan.  1. 

Problem  2.  — What  is  the  balance  due  July  1,  on  the  fol- 
lowing account  ?  Deposits  :  Nov.  20,  $300  ;  Jan.  14,  $200  ; 
June  10,  $150.  Withdrawals:  March  20,  $150;  June  20, 
$100.  Interest  is  declared  every  6  months,  and  3%  per 
annum  is  allowed  from  the  first  of  each  quarter. 


Dates 

Deposits 

"WiTIIDEAWALS 

Daily 

Balances 

Interest 
Days 

Smallest 
Quarterly 
Balances 

Qttarterly 
Interest 

Nov.  20 
Jan.  14 
Mar.  20 
April  1 
June  10 
June  20 

$300 
200 

150 

$150 

100 

$300 
500 
350 
350 
500 
400 

Jan.    1 

April  1 
July    1 

$350 

400 

$5.25 

6.00 

11.25 

5.63 

5.62 

400.00 

$405.62 

The  smallest  balance  for  the  first  quarter  is  $350,  and  for 
the  second  quarter,  $400.  The  quarterly  interest  on  these 
balances  aggregates  $5.62.  $405.62  is  the  balance  due 
July  1. 

PROBLEMS 

1.  A.  B.  Wilson  made  the  following  deposits  in  a  savings  bank :  Dec. 
20,  $60;  June  30,  $80;  Sept.  1,$100;  Oct.  15,  $200;  Nov.  5,  $100;  Dec. 
25,  $200.  The  interest  term  is  6  months,  and  interest  is  allowed  on 
balances  from  the  first  of  each  quarter  at  4  %  per  annum.  What  is  the 
balance  due  Jan.  1  ? 

Note.  —  The  interest  is  compounded  at  end  of  each  interest  term. 


BANKING  AND   DISCOUNT  209 

2.  A.  G.  Thomas  made  the  following  deposits  in  a  savings  bank : 
Dec.  15,  1^300;  Jan.  14,  $300;  Feb.  25,  $150;  June  5,  $100;  July  1, 
$  120.  The  interest  term  is  3  months,  and  interest  at  the  rate  of  3%  is 
computed  from  the  first  day  of  each  quarter.  What  amount  is  due 
Julyl? 

3.  A.  M.  Peabody  deposits  in  a  savings  bank  as  follows :  Jan.  1,  $400 ; 
Feb.  20,  $200;  March  10,  $150;  April  10,  $60;  May  15,  $5.50.  He 
withdrew  during  the  same  time,  as  follows  :  Jan.  15,  $  100 ;  Feb,  5,  $150 ; 
April  20,  $  80 ;  June  30,  $  120.  The  rate  of  interest  is  4  %  per  annum,  the 
interest  term  6  months,  and  the  interest  is  computed  from  the  first  of 
each  quarter.     Find  the  amount  due  July  1. 

4.  A  customer  makes  deposits  in  a  savings  bank  as  follows  :  Nov.  25, 
$  600 ;  Jan.  1,  $  200;  May  20,  $  50 ;  June  25,  $  100.  If  the  interest  term 
is  6  months,  and  interest  is  computed  from  the  first  day  of  each  month 
at  4%,  what  is  due  July  1  ? 

NEGOTIABLE  PAPER 

182.  Checks.  Money  on  deposit  in  a  bank  in  an  "open 
account"  is  subject  to  check.  That  is,  the  bank  will  pay 
out  any  part  or  all  of  it  upon  a  written  order  from  the  de- 
positor. This  order  is  called  a  cJiech,  Checks  are  usually 
made  payable  to  a  certain  person  known  as  the  payee^  or  to 
his  order,  although  sometimes  they  are  made  payable  to  the 
bearer.  When  the  check  is  paid  by  the  bank,  it  is  stamped 
"  paid  "  and  is  finally  returned  to  the  depositor  who  signed 
the  check,  known  as  the  maker.  To  withdraw  money  from 
deposit,  the  check  is  made  payable  to  "self." 

If  there  are  no  funds  on  deposit,  a  check  is  worthless.  Ob- 
taining money  from  a  third  party  on  such  a  check  is  against 
the  law. 

For  certain  purposes,  it  is  desirable  to  have  a  check  show 
on  its  face  that  it  represents  actual  value.  To  do  this,  the 
maker  or  payee  takes  it  to  the  bank  upon  which  it  is  drawn, 
and  the  paying  teller  or  cashier  of  the  bank  writes  across  its 
face,  and  over  his  signature,  that  it  is  "  good  when  properly 
indorsed."      This  makes  of  it  a  certified   cheek.      Enough 


210  ELEMENTS  OF  BUSINESS  ARITHMETIC 

money  belonging  to  the  maker  is  set  aside  by  the  bank  to 
pay  it  when  presented.  This  amount  will  be  paid,  or  re- 
leased for  regular  account,  only  upon  presentation  of  the 
check  itself. 

Checks  signed  by  individual  depositors  of  a  bank  are  known 
as  individual  eheclcs.  Those  signed  by  the  cashier  of  a  bank 
are  known  as  cashier's  cheeks.  The  latter  are  used  in  paying 
bank  expenses,  or  to  pay  the  proceeds  of  a  note  purchased, 
and  are  sometimes  issued  to  customers  to  be  used  instead  of 
drafts. 

183.  Certificates  of  Deposit.  Money  may  be  deposited  for 
special  purposes,  in  a  "closed  account,"  i.e.  not  subject  to 
check.  The  bank  issues  a  certificate  of  deposit,  therefore, 
certifying  that  a  certain  sum  is  on  deposit,  which  will  be 
paid  to  the  holder  of  the  certificate  when  properly  indorsed. 
This  may  be  either  a  demand  or  a  time  certificate.  Time  cer- 
tificates usually  draw  interest. 

184.  Notes.  When  promissory  notes  (Sec.  161)  are  made 
payable  to  a  particular  person,  without  having  either  of  the 
two  clauses,  "  or  order  "  or  "  or  bearer,"  they  are  payable  to 
no  person  other  than  the  payee,  and  are  therefore  not  trans- 
ferable. Such  notes  are  non-negotiahle.  If  either  of  the 
clauses  are  embodied,  the  note  may  be  bought  and  sold,  and 
it  is,  therefore,  negotiable.  Checks,  notes,  drafts,  certificates 
of  deposit,  or  any  other  papers  representing  value,  which 
permit  of  being  bought  and  sold,  are  likewise  termed  nego- 
tiable papers. 

185.  Exchange.  Bank  Drafts.  The  use  of  bank  drafts  as 
a  means  of  exchange  has  already  been  treated  in  Sec.  118. 
As  there  stated,  a  bank  usually  maintains  credit  with  an- 
other bank,  its  correspondent,  in  some  of  the  larger  cities 
and  sells  drafts  on  that  credit  for  a  certain  fixed  charge  for 


BANKING  AND  DISCOUNT  211 

small  amounts,   and  at  so  much   per  hundred   dollars  for 
larger  amounts. 

Sometimes  the  amount  charged  for  exchange  is  expressed 
in  per  cents,  and  when  so,  it  is  fixed  variously  at  between 
Jq  %  and  ^  %  of  the  amount  sold.  The  price  would  depend 
upon  the  size  of  the  draft  and  sometimes  upon  the  demand 
for  exchange  on  the  particular  city  in  which  the  paying 
bank  is  located.  Formerly  it  was  the  custom  to  quote  ex- 
change at  either  a  premium  or  a  discount,  dependent  upon 
whether  the  amount  of  credit  on  a  given  city  was  greater  or 
less  than  the  demands  of  business.  Of  late  years,  this  finds 
very  little  use,  and  drafts  are  sold  at  par,  the  only  charge 
being  that  for  the  service. 

186.  Collection  by  Draft.  Individual  drafts  form  a  very 
common  means  of  collecting  delinquent  accounts.  If  a 
debtor  has  paid  no  attention  to  repeated  statements  or  let- 
ters, he  is  notified  that  unless  heard  from  by  a  certain  date, 
he  will  be  drawn  upon  for  the  amount  of  the  debt. 

If  no  word  comes,  a  draft  is  drawn  on  him  in  favor  of  the 
creditor,  and  made  payable  "at  sight,"  or  some  number  of 
days  after  sight.  This  is  usually  deposited  in  the  bank  with 
which  the  creditor  does  business,  and  it  is  sent  to  a  bank 
in  the  city  of  the  debtor's  residence  indorsed  "for  collec- 
tion," and  the  latter  bank  will  present  it  and  endeavor  to 
have  it  paid.  If  successful,  the  bank  will  remit  the  amount, 
less  a  small  fee  for  the  service.  If  the  draft  be  not  paid, 
it  will  be  returned  with  a  memorandum  giving  the  reason, 
should  one  be  given  by  the  debtor,  for  its  nonpayment. 
Repeated  refusal  or  failure  to  pay  such  drafts  leaves  one 
open  to  suspicion  of  unreliability. 

If  the  draft  is  for  a  period  of  time  after  sight,  the  drawee 
who  wishes  to  honor  the  draft  writes  the  word  accepted^ 
with  date  and  signature,  across  the  face  of  the  draft,  which 


212  ELEMENTS  OF  BUSINESS  ARITHMETIC 

thereupon  to  all  intents  and  purposes  becomes  a  promissory 
note,  and  .the  time  period  is  counted  from  the  date  of  the 
acceptance.  If  a  draft  reads  "after  date,"  then  the  time 
period  begins  with  the  date  of  the  draft. 

The  signer  of  a  draft  is  the  drawer^  the  person  or  bank  on 
whom  it  is  drawn  is  the  drawee^  and  the  person  to  whom  it  is 
to  be  paid  is  the  payee. 

In  much  the  same  way  shippers  may  use  individual  drafts 
to  secure  payment  on  a  shipment  of  goods  to  an  unknown  or 
unreliable  customer.  The  goods  are  billed  to  the  shipper  at 
the  address  of  the  customer,  and  the  bill  of  lading,  together 
with  the  draft  for  the  amount  of  the  invoice,  is  sent  to  a 
bank  for  collection.  Upon  payment  of  the  draft,  the  cus- 
tomer is  given  the  bill  of  lading,  which  entitles  him  to  the 
delivery  of  the  goods  shipped. 

187.    Indorsement.     When  one  wishes  to  cash  a  check,  he 

writes  his  name  on  the  back  of  the  check  and  presents  it 

for  payment.      This  is  known  t>i     i    •  j  ^ 

^  -^  1  .        P  Blank  indorsement 

as  an  indorsement^  and  its  ef-  A    M    H*     k' 

feet  is  to  transfer  ownership  ry    /•    i      -  j  * 

^         Particular  indorsement 
of  the  check,  or  the  money  it     p      ^^  ^^^^^  ^^^^^^^^^  ^^_ 

represents,   to   whoever    may  j^^^^  -^^^^^^ 

hold  the  check. 

When  one  wishes  to  sell  a  negotiable  note,  the  payee,  like- 
wise, indorses  it  upon  the  back.  This  may  be  done  "in 
blank"    by    simply    writing    his  ^^u  indorsement 

name,   when   it   becomes    payable        p^^  ^^  ^^^  ^^^^^  ^^ 
to  the  bearer  of  the  note;   or  it  ^^^^^^  &  Sons 

may  be  indorsed  "  in  particular  "  ^    ^    Dawson 

by   writing    "pay   to " 

and  signing  it,  when  it  is  payable  only  to  the  one  whose  name 
is  written ;  or  it  may  be  indorsed  "  in  full,"  by  writing  "  pay 
to  the  order  of "  and  signing  it.     This  latter  in- 


BANKING  AND  DISCOUNT 


217 


V      Find  the  date  of  maturity  of  the  following  drafts ; 


Date  Accepted 

Time 

Date  Accepted 

Time 

5.   May  9 

20  da. 

7.  Jan.  30 

30  da. 

6.  June  12 

1  mo. 

8.   Sept.  6 

3  mo. 

Find  date  of  maturity  and  term  of  discount: 

Date  of  Note 

1                              Time 

Date  of  Discount 

9.  May  15 

3  mo. 

June  1 

10.   Sept.  20 

30  da. 

Sept.  29 

11.   March  18 

2  mo. 

March  31 

Date  of  Draft      Time  after  Sight 

When  Accepted 

"When  Discounted 

12.    Feb.  16 

10  da. 

Feb.  17 

Feb.  18 

13.  March  20 

30  da. 

March  22 

March  25 

14.   Julyl 

60  da. 

July  5 

July  20 

Date  of  Draft      Time  after  Date 

When  Accepted 

When  Discounted 

15.   July  6 

3  mo. 

July  10 

Aug.  4 

16.    Aug.  8 

1  mo. 

Aug.  9 

Aug.  9 

17.   Sept.  15 

30  da. 

Sept.  15 

Sept.  18 

PROBLEMS 

Find  bdnk  discount  and  proceeds: 

Face  .         Date  of  Note        Time 

Date  of  Discount 

Rate  of  Discount 

1.  $1200 

June  2          30  da. 

June  6 

6% 

2.  12500 

April  8         60  da. 

April  20 

6% 

3.  $3000 

Aug.  10          2  mo. 

Aug.  15 

5% 

4.  $   600 

Sept.  16          2  mo. 

Sept.  20 

8% 

5.  17200 

Jan.  14         90  da. 

Jan.  25 

^% 

6.  $3500 

May  20        30  da. 

May  21 

7% 

7.  May  15,  A.  B.  Kittridge  &  Co.  borrowed  of  the   First   National 
Bank  $1600  on  their  note  at  60  days.     Find  the  proceeds,  the  rate  being 

7%. 

8.  I  borrowed  $450  of  the  Atlas  Bank  on  my  note  for  70  days. 
Write  the  note,  and  find  the  proceeds,  the  rate  of  discount  being  6  %. 


218 


ELEMENTS   OF  BUSINESS  ARITHMETIC 


Find  the  date  of  maturity,  the  term  of  discount,  the  discount,  and  the 
proceeds  of  the  following  notes  and  drafts : 

9.   $1800.00  Columbus,  O.,  April  12. 

Sixty  days  after  date  I  promise  to  pay  to  the  order  of  O.  E.  Chase  & 
Sons,  Eighteen  Hundred  Dollars,  at  the  Commercial  National  Bank. 
Value  received. 

Discounted  May  8,  at  5%.  A.  B.  Grindle. 

10.  $660.00  Minneapolis,  Minn.,  Sept.  1. 
Three  months  after  date  I  promise  to  pay  to  the  order  of  Freeman  & 

Comstock,  Six  Hundred  Sixty  Dollars,  at  the  Bank  of  Commerce.     Value 
received. 

Discounted  Oct.  10,  at  4^%.  C.  M.  Dunlap. 

11.  $850.00  Cleveland,  O.,  May  1,  19—. 
Six  months  after  date  I  promise  to  pay  to  the  order  of  A.  Douglas  & 

Co.,  Eight  Hundred  Fifty  Dollars,  with  interest  at  5  %.     Value  received. 
Discounted  June  6,  at  6  %.  C.  L.  Trueblood. 

12.  $875.00  Lead,  S.  Dak.,  May  29,  19—. 
Eight  months  after  date  I  promise  to  pay  to  the  order  of  L.  M.  Gittin- 

ger,  Eight   Hundred  Seventy-five  Dollars,  with  interest  at  6%.     Value 
received. 

Discounted  June  25,  at  7%.  Oliver  C.  Ditson. 

13.  $2750.00                             Kansas  City,  Mo.,  March  13,  19—. 
At  sixty  days'  sight  pay  to  the  order  of  ourselves 

Twenty-seven  Hundred  Fifty  §-g- Dollars. 

Value  received,  and  charge  the  same  to  the  account  of 

To  Ensign  Bros.,  G.  W.  Patchell  &  Co. 

Sandusky,  O. 

Accepted  March  30.     Discounted  April  4,  at  7  %. 

14. 


^^l'6^~^^^:^/i^^i^'y?^^i^'^ 


BANKING  AND  DISCOUNT  219 

15. 


16.  $675.50  Cincinnati,  O.,  Dec.  23,  19—. 
Ninety  days  after  date  pay  to  the  order  of  ourselves.  Six  Hundred 

Seventy-five   and  -^^^    Dollars.      Value    received,   and    charge    to    the 
account  of 

To  A.  B.  HiMES,  Antwerp  &  Bragg. 

Indianapolis,  Ind. 

Accepted  Jan.  2.     Discounted  Jan.  4,  at  7%.     Collection  charges  xV%« 

Note.  —  Collection  is  charged  on  the  face. 

17.  I  wish  to  borrow  $600  at  the  bank.  For  what  sum  must  I  issue 
a  60-day  note  to  obtain  the  amount,  discount  being  6%? 

18.  I  owe  $960,  and  have  my  note  discounted  at  the  bank  for  75  days 
at  6%  iov  such  a  sum  that  the  proceeds  will  pay  the  debt.  What  was  the 
face  of  the  note  ? 

19.  A  merchant  purchased  goods  for  $875  on  3  months'  credit,  5%  be- 
ing offered  him  for  cash.  He  accepted  the  cash  offer  and  borrowed  the 
money  at  the  bank,  giving  his  note  for  60  days  at  6%.  What  was  the 
face  of  the  note,  and  what  did  he  gain  or  lose  by  so  doing? 

20.  Abankdraftfor$7500  was  bought  for  $7496.25.  What  was  the 
rate  of  exchange  ?  At  the  same  rate,  what  would  be  the  cost  of  a  draft 
for  $14,500?  one  for  $125,455.60?  one  for  $12,367.50? 

21.  An  agent  sold  a  carload  of  26  cattle,  averaging  1125  lb.,  at  $5.60 
per  hundred  weight.  He  paid  $  135  freight,  $  26.75  for  feed,  and  charged 
2  %  commission  for  selling.  He  buys  a  draft  at  ^  %  preinium  with  the 
proceeds.     What  is  the  face  of  the  draft? 

22.  Snyder  &  Co.  of  New  Orleans  drew  a  draft  on  A.  M.  Hawkins  of 
Boston,  Mass.,  for  $7865.50,  which  they  sold  at  the  bank  at  f%  discount. 
What  M'^ere  the  proceeds  ? 


220 


ELEMENTS   OF  BUSINESS  ARITHMETIC 


23.   Complete  the  following  letter  of  advice.     The  rate  of  collection 
on  the  first  two  items  is  ^^%,  and  on  the  others  \%. 


SECOND  NATIONAL  BANK 

Richmond,  Ind.,  June  20,  1917. 
Mr.  WM.  J.  DOYLE,  Cashier, 
Atlas  Bank, 

Cincinnati,  Ohio. 
Dear  Sir: 

We  credit  your  account  this  day  for  the  proceeds  of  collec- 
tions as  stated  below. 

Respectfully  yours, 

JAMES  W.  KING,  Cashier. 

YOITE  No. 

Payer 

Amount 

Charges 

Peoceeds 

620 
415 
930 
560 

748 

A.  M.  Pierson 
F.  T.  Davis 
Murphy,  Grant  &  Co. 
Richmond  Chemical  Co. 
S.  F.  Carroll 

600 

560 

3545 

12345 

800 

00 
75 
10 
80 
00 

XIX 

STOCKS  AND  BONDS 

196.  Organizations  for  Business.  If  an  individual  engages 
in  business  by  himself,  he  is  entitled  to  all  the  profits  and 
assumes  personal  liability  for  all  debts.  Should  it  be  de- 
sirable to  have  the  capital  or  services  of  more  than  one  per- 
son, a  partnership  is  formed.  The  business  is  then  done 
under  a  firm  name,  e.g.  Merritt  &  Saunders,  John  N.  Bald- 
win &  Co.,  Parlin,  Orendorff  &  Co.,  etc. 

There  is  usually  a  written  agreement  for  such  a  partner- 
ship, specifying  the  services,  money,  or  property  contrib- 
uted by  each  to  the  business,  and  stating  what  part  of  the 
profits  each  is  to  receive.  Each  partner  may  bind  the 
firm  by  his  acts,  and  each  is  personally  liable  for  the  firm's 
indebtedness. 

197.  Corporations.  Whenever  large  capital  is  needed  for 
a  business,  or  investors  wish  to  limit  their  financial  responsi- 
bility to  the  amount  invested,  or  when  the  range  of  the 
business  is  wide,  or  individuals  wish  to  invest  in,  but  not  to 
give  their  personal  attention  to  a  business,  and  for  various 
other  reasons,  a  stock  company  or  corporation  is  organized. 

Such  an  organization  is  usually  created  under,  and  must 
conform  to,  state  laws.  Its  affairs  are  conducted  by  officers, 
selected  in  a  prescribed  way.  When  organized  it  becomes 
before  the  law  a  body  corporate.,  vested  with  the  same  rights 
as  an  individual ;  to  have  and  to  hold  property,  to  contract 
debts  (within  the  limits  of  the  law),  to  sue  and  be  sued,  etc. 

221 


222  ELEMENTS  OF  BUSINESS  ARITHMETIC 

198.  Articles  of  Incorporation.  In  order  to  form  a  corpo- 
ration, the  investors  or  incorporators  sign,  file  in  some  gov- 
ernment office,  and  publish  their  Articles  of  Incorporation. 
These  articles  set  forth  the  purposes  of  the  organization,  the 
amount  of  money  subscribed  by  each,  the  capital  stocky  the 
total  amount  that  may  be  subscribed  or  authorized  capital^ 
the  number  of  shares  or  parts  into  which  the  capital  is  to  be 
divided,  the  face  or  par  value  of  each  share,  the  name  of  the 
company,  its  place  of  business,  its  officers,  etc. 

199.  Certificates  of  Stock.  When  the  legal  requirements 
are  satisfied,  the  subscribers  pay  into  the  treasury  the  stipu- 
lated price  and  receive  a  certificate  of  stocky  stating  the  num- 
ber of  shares  bought,  the  par  value  of  each  share,  etc.  The 
owner  of  such  certificate  is  entitled  to  a  part  in  the  control 
of  the  business  corporation,  and  to  participate  in  its  property 
and  its  profits  in  proportion  to  the  number  of  shares  owned. 

200.  Dividends  and  Assessments.  At  stated  periods,  the 
condition  of  the  business  is  ascertained.  If  profits  are 
shown,  a  portion  is  distributed  among  the  stockholders  as  a 
dividend.     Gains  remaining  are  termed  undivided  profits. 

If  a  loss  is  shown,  which  the  necessities  of  the  business 
require  should  be  made  good,  it  is  apportioned  among  the 
stockholders  as  an  assessment  to  be  paid  by  them.  Dividends 
and  assessments  are  usually  expressed  in  per  cents  of  the 
face  value  of  the  stock. 

202.  Kinds  of  Stock.  Corporations  often  issue  both  pre- 
ferred and  common  stock.  Dividends  up  to  a  certain  limit, 
usually  5  %  to  7  %,  are  first  paid  on  preferred  stock.  Profits 
remaining  may  then  be  divided  on  common  stock.  Holders 
of  preferred  stock  thus  have  first  chance  for  dividends,  but 
their  dividends  are  limited.  Law  or  charter  also  sometimes 
limits  dividends  on  common  stock. 


STOCKS  AND  BONDS  223 

202.  Premium  and  Discount.  When  dividends  declared 
by  a  given  concern  are  higher  than  the  prevailing  rates  of 
interest,  the  stock  of  such  a  company  will  naturally  sell  for 
more  than  the  face  value  of  the  shares.  It  is  then  said  to 
be  above  par^  or  at  a  premium.  Should  the  dividends  be  less 
than  current  interest  rates,  the  stock  will  not  bring  its  face 
value  and  is  said  to  be  below  par^  or  at  a  discount. 

The  market  value  of  stock  is  the  amount  it  will  bring  in 
the  market,  and  is  usually  stated  as  a  per  cent  of  the  par 
value.  Thus,  stock  quoted  at  26,  84,  or  120  means  that  a 
hundred  dollars  in  stock  of  a  company  is  worth  |26,  184,  or 
1120,  respectively. 

203.  Liability  of  Stockholders.  In  general,  a  holder  of 
stock  in  a  business  corporation  is  liable  for  the  debts  of  the 
corporation  only  to  the  extent  of  the  par  value  of  the  stock. 
The  National  Banking  Law,  however,  makes  stockholders  in 
national  banks  liable  to  the  extent  of  the  par  value  in  addi- 
tion to  what  they  have  paid  for  the  stock. 

Some  stock  is,  by  the  terms  of  the  charter  or  the  by-laws, 
non-assessable.  In  such  companies,  the  entire  risk  assumed 
by  the  holder  of  the  stock  is  the  amount  paid  for  it. 

204.  Bonds  of  Corporations.  Corporations  may  borrow 
money,  pledging  their  property  as  security,  in  the  same  way 
as  individuals.  If  some  particular  property  is  pledged,  it  is 
upon  an  ordinary  promissory  note,  with  real  or  chattel 
mortgage. 

When  the  amount  of  money  to  be  raised  is  large,  it  is 
done  by  formally  issuing  bonds.,  and  placing  them  upon  the 
market  for  sale.  A  bond,  then,  is  a  mortgage  note,  upon 
which  the  corporation  pays  interest,  and  to  the  payment  of 
which  the  entire  property  and  business  of  the  corporation  is 
pledged. 


224  ELEMENTS  OF  BUSINESS  APITHMETIC 

205.  Government  Bonds.  National  and  state  governments, 
counties,  townships,  cities,  school  districts,  etc.,  are  by  law 
declared  to  be  "bodies  corporate."  As  such  they  may  issue 
bonds,  within  certain  limits,  and  agree  to  pay  a  limited  rate 
of  interest.  For  the  redemption  of  such  bonds  a  tax  is 
levied  to  create  a  sinking  fund  for  their  payment  when  due. 
In  default  of  payment,  the  courts  may  issue  judgment  and 
cause  a  special  tax  to  be  levied  and  collected  for  their  pay- 
ment. 

206.  Kinds  of  Bonds.  When  bonds  are  made  payable  to 
the  owner  or  his  assignee,  they  are  termed  registered  bonds. 
The  names  of  owners  are  registered,  and  the  interest  is  sent 
directly  to  them. 

When  the  bonds  are  made  payable  to  bearer,  the  interest 
is  provided  for  in  attached  notes,  termed  coupons.  A 
coupon  is  surrendered  when  each  interest  payment  is  made. 
Bonds  of  this  sort  are  termed  coupon  bonds. 

207.  Bond  Values.  The  value  of  a  business  corporation 
bond  depends  upon  the  amount  of  property  or  business  of 
the  company.  If  the  property  is  large  or  the  business  pros- 
perous, then  the  bonds  are  reliable.  Their  market  value 
also  depends  upon  the  rate  of  interest  the  bonds  bear.  If  it 
is  higher  than  the  current  interest  rates,  the  bonds  form  a 
profitable  investment,  and  will,  therefore,  tend  to  sell  at  a 
premium.  If  the  interest  rate  is  less  than  current  rates,  or  if 
the  bonds  are  not  absolutely  secure,  they  will  probably  sell 
at  a  discount. 

208.  Stock  Quotations.  The  price  of  stocks  or  bonds  is 
usually  quoted  at  a  certain  per  cent  of  their  par  value.  So 
many  elements  enter  into  the  question  of  the  value  of  stocks 
and  bonds  in  the  open  market,  and  so  easily  is  the  confidence 
of    the   investing    public   weakened   or   strengthened,  that 


STOCKS  AND  BONDS  225 

the  market  quotations  of   stocks  or  bonds  often  fluctuate 
widely. 

The  necessity  for  an  intimate  knowledge  of  corporation 
and  market  conditions  gives  rise  to  brokerage  firms.  These 
brokers  advise  their  clients,  buy  and  sell  stocks,  bonds,  etc., 
and  charge  a  small  commission  or  brokerage^  usually  about 
\  of  one  per  cent,  for  the  service.  Brokerage  is  always  a  per 
cent  of  the  par  value,  whether  for  buying  or  for  selling. 

209.  Stock  Exchanges.  So  important  are  stocks  and  bonds 
that  special  organizations  of  dealers  are  formed,  known  as 
stock  exchanges,  boards  of  trade,  etc.  At  these  exchanges, 
brokers  buy  and  sell  for  investors.  If  for  speculation,  in- 
vestors usually  buy  because  of  an  expected  rise  in  the  market 
value,  expecting  to  sell  at  an  advanced  price.  They  do  not 
always  pay  the  full  price  of  the  stock,  but  a  part  only,  leav- 
ing the  certificate  of  stock  with  the  broker  as  security  for 
the  remainder.  Thus,  stock  may  be  bought  upon  a  20  % 
margin,  by  paying  20  %  of  its  value,  depending  on  profits 
to  pay  the  remainder  or  using  the  amount  paid  to  cover 
losses  if  the  stock  goes  down. 

There  is  usually  a  group  of  operators^  who  are  interested 
in  forcing  the  price  of  certain  stocks  upward.  These  are 
known  as  bulls.  There  are  others  who  wish  to  see  the  price 
lowered,  and  these  are  called  bears.  Bears  may  either  wish 
to  purchase  good  stock  at  a  cheap  price,  or  in  the  belief  that 
the  stock  was  certain  to  go  down,  they  may  have  sold  largely 
of  the  stock  without  owning  it,  and  wish,  therefore,  to  force 
it  down  so  they  may  purchase  cheaply  what  they  have  bar- 
gained to  deliver.  In  the  latter  case  they  are  said  to  have 
sold  short. 

210.  Market  Quotations.  The  following  quotations  show 
the  highest  and  the  lowest  market  quotations  for  a  given 
year  as  furnished  by  Bradstreet's  Commercial  Agency : 


226  ELEMENTS  OF  BUSINESS  ARITHMETIC 

Stocks 

High  Low 

Adams  Express 250  236 

Amalgamated  Copper 91  70 

American  Beet  Sugar 34^  23 

American  Express 246  209^ 

Atchison,  Topeka  &  Santa  Fe      .     .    .  93|  71J 

Brooklyn  Rapid  Transit 91 1  56| 

Chicago,  Burlington  &  Quincy    .     .     .  250  201 

Chicago,  Milwaukee  &  St.  Paul  .     .     .  187i  168^ 

Consolidated  Coal 73  24| 

Erie 52|  37| 

General  Chemical 72|  37i 

Illinois  Central 183  152| 

National  Biscuit 66f  52 

National  Biscuit  Pfd 120|  110 

National  Lead 77^  24| 

Pittsburg,  Ft.  Wayne  &  Chicago     .     .  185  182^ 

Pressed  Steel  Car 53|  34 

Quicksilver 1|              | 

Rubber  Goods  Mfg.  Co 39  25 

Union  Pacific 138f  113 

United  States  Leather 16  10| 

United  States  Steel 39^  24| 

Bonds 

High  Low 

Am.  Hide  &  Leather  6's 97^  97| 

Chesapeake  &  Ohio  6's,  1911  ....  110  110 

Denver  &  Rio  Grande  4's 100|  97^ 

Illinois  Central  4's,  1952 108  107^ 

Missouri  Pacific  4's 96  95 

Seaboard  Air  Line  5's 104^  104 

United  States  reg.  4's,  1907     ....  104^  104| 

Pennsylvania  4|'s 109  108^ 

United  States  coupon  4's 104^  104 

211.  Illustrative  Problems.  In  all  problems  in  this  text, 
the  par  value  of  a  share  of  stock  will  be  taken  as  f  100,  unless 
otherwise  stated. 


STOCKS  AND  BONDS  v  227 

1.  A  broker  sells  145  shares  of  stock  at  125| ;  brokerage 
■|%.     What  should  Im  principal  receive? 

125|  %  -  1  %  =  125|  %,  proceeds  of  each  share. 

f  100  X  145  =  114,500,  par  value  of  145  shares. 

114,500  X  1,25J  =  118,161.25,  proceeds.  '^,; 

II 

2.  A  broker  sold  Union  Gas  stock  for  126,250  at  75% 
premium.  How  many  shares  did  he  sell?  What  was  the 
par  value  of  the  stock? 

I  =  selling  price.  ^\'\*>Y 
Jof  par  value  =  126,250. 

^  =  |of  126,250,  or  13750. 
I  =  $3750  X  4,  or  $15,000,  par  value  of  stock. 
$15,000  -^  $100  =  150,  number  of  shares.    ,  v 

3.  What  sum  must  be  invested  in  6  %  bonds  at  120  to 
yield  an  annual  income  of  %  2820  ? 

6  %  of  face  value  of  bonds  bo,ught  =  $2820.;^ 
100  %  =  -ig^  of  $2820  or  |47,0i)0  face  value. 
120  %=f.  ' 

I  of  $47,000  =  $56,400,  investment. 

4.  What  per  cent  profit  does  an  investor  make  on  stock 
that  pays  a  dividend  of  6  %,  if  he  buys  at  75  ? 

$  6  =  income  on  one  share.  /  v^ 

$75  =  cost  of  that  share. 
A  =  8%,  profit.  N 

5.  A  year's  net  profits  of  the  Plymouth  Milling  Co.  were 
$15,275.50.  The  capitalization  of  the  concern  is  $200,000, 
divided  into  2000  shares.  A  dividend  of  6|  %  was  declared, 
and  the  remainder  of  the  profits  was  carried  to  surplus  fund. 
Find  the  amount  of  dividend  and  the  amount  carried  to  sur- 


228  ELEMENTS  OF  BUSINESS  ARITHMETIC 

plus  fund.     What  amount  will  a  man  receive  who  owns  60 
shares  of  stock? 

6-1-  %  of  1200,000  =  113,000,  the  dividend  declared. 
115,675.50  -  f  13,000  =  $2675.50,  surplus  fund. 

6i  %  of  $6000  =  1390,  dividend  on  60  shares  of  stock. 

6.  A  manufacturing  company  is  capitalized  at  $200,000. 
The  gross  earnings  for  a  year  are  $25,185,  and  the  expenses 
are  $6785.50.  After  setting  aside  2%  for  surplus  fund, 
what  even  per  cent  of  dividend  may  be  declared  ? 

$25,185  -  $6785.50  =  $18,399.50,  net  earnings. 
2  ojo  of  $18,399.50  =  $367.99,  amount  for  surplus  fund. 
$18,399.50  -  $367.50  =  $18,031.50,  amount  to  be  divided. 
1%  of  $200,000  =  $2000,  amount  of  1%  dividend. 
$18,031.51  ^  $2000  =  9,  the  rate  per  cent  of  dividend,  with 
$31.51  additional,  carried  to  undivided  profits. 

PROBLEMS 

Find  the  market  value  at  the  highest  and  lowest  price  of  the  following 
stocks  and  bonds,  by  use  of  the  above  market  quotations  : 

1.  75  shares  Adams  Express.  3.  155  shares  National  Lead. 

2.  68  shares  Erie.  4.   85  shares  Quicksilver. 

5.  150  shares  Union  Pacific. 

6.  7  Pennsylvania  4|'s  (Denom.  ^1000). 

7.  9  Am.  Hide  and  Leather  6's  (Denom.  $1000). 

8.  245  shares  National  Biscuit  pfd. 

9.  A  man  who  holds  170  shares  of  stock  receives  a  dividend  of  $1275. 
What  was  the  rate  of  dividend  declared  ? 

10.  An  assessment  of  $306  is  made  on  72  shares  of  mining  stock. 
What  is  the  rate  of  assessment  ? 

>  11.   A  broker  sold  for  me  360  shares  of  gas  stock  at  145 ;  brokerage  \%. 
What  sum  should  I  receive? 

12.  A  railroad  declares  a  dividend  of  5%.  How  many  shares  does  a 
man  hold  who  receives  a  dividend  of  $435,  if  the  par  value  of  stock  is 
$100? 


STOCKS  AND  BONDS  229 

^13.   What  sum  must  be  sent  a  broker  that  he  may  buy  150  shares  of 
gas  stock  at  105;  brokerage  |%? 

14.  A  stockholder  meets  an  assessment  of  $167.50,  which  is  levied  at 
2^  %  on  his  stock.     How  many  shares  has  he  ? 

15.  How  many  shares  of  Atchison,  Topeka,  and  Santa  Fe  stock  can 
be  bought  for  |8100  at  89|;  brokerage  ^%?  What  will  the  dividend  on 
this  stock  amount  to  at  5%?  What  rate  of  interest  would  this  be  on  the 
investment  ? 

16.  What  amount  of  stock  must  be  sold  at  41^  to  yield  $8275,  if 
brokerage  is  i%? 

17.  How  many  shares  of  American  Sugar  at  31 1  can  be  bought  for 
$202,400;  brokerage  i%?  If  this  stock  pays  a  dividend  of  2|%,  what 
rate  of  interest  will  a  man  receive  on  his  investment? 

18.  What  sum  must  be  invested  at  93  to  bring  an  income  of  $4800,  if  j 
the  rate  of  dividend  is  4%'^ 

19.  What  is  the  quotation  of  7%  stock  that  brings  an  income  of  10%? 

20.  Wliat  rate  per  cent  is  realized  on  an  investment  by  investing  in 
5%  stock  at  80? 

21.  A  banker  sold  through  a  broker  150  shares  of  stock  at  124|,  pay- 
ijQg  i%  brokerage.     What  amount  did  each  receive? 

22.  A  speculator  bought  2500  shares  of  the  United  States  Steel  stock 
at  33|  and  sold  it  at  39f.  What  was  his  net  profit  after  allowing  \% 
brokerage  each  for  buying  and  selling  ? 

23.  A  man  buys  120  shares  of  stock  at  76^  and  six  months  later  sells 
it  for  85.  In  that  time  he  received  a  dividend  of  lf%.  If  money  is 
worth  6%  interest  and  he  paid  \%  brokerage  for  buying  and  for  selling, 
did  he  gain  or  lose,  and  how  much? 

24.  The  capital  stock  of  a  company  is  $200,000.     \  of  this  is  preferred 
stock,  entitled  to  6%  dividend.     What  rate  of  dividend  is  paid  on  com-  - 
mon  stock,  if  $9000  is  distributed  in  dividends? 

25.  An  investor  buys  604  shares  of  stock,  par  value  $  50,  for  $  35  a 
share;  brokerage  \%.     Six  months  later  he  sells  for  $58  a  share.     In  the    * 
meantime  he  had  received  a  dividend  of  5%.     Money  being  worth  6%, 
what  did  he  gain  or  lose? 

26.  A  bank  with  capital  stock  of  $150,000,  declares  a  semi-annual  j 
dividend  of  4^%.  What  is  the  amount  of  the  dividend,  and  how  muchy 
will  a  man  receive  annually  who  owns  275  shares?  ^    / 


230  ELEMENTS  OF  BUSINESS  ARITHMETIC 

27.  A  corporation  has  a  capital  stock  of  $100,000.  Its  net  earnings 
for  the  year  are  114,256.32.  4  %  of  the  net  earnings  is  set  aside  as  a  sur- 
plus fund  to  cover  losses,  7  %  dividend  is  declared,  and  the  remainder  is 
carried  to  undivided  profits.  What  are  the  amounts  carried  to  surplus 
fund,  undivided  profits,  and  to  dividend  accounts? 

28.  A  merchant  sold  his  business  for  $245,000.  He  invested  $196,000 
in  Pullman  stock  at  195|,  and  the  remainder  in  Erie  preferred  at  69 J. 
Pullman  stock  pays  12%  and  Erie  6%  dividend;  brokerage  ^%  in  each 
case  for  buying.     What  was  his  annual  income  ? 

29.  A  broker  purchased  500  shares  Amalgamated  Copper  at  70 ;  250 
shares  C.  B.  &  Q.  at  201;  400  shares  National  Biscuit  Co.  at  66| ;  475 
shares  General  Chemical  at  72|,  and  150  shares  Western  Union  at  113. 
What  is  the  total  cost  to  his  principal,  if  brokerage  is  ^%? 

30.  A  speculator  purchased  200  shares  United  States  Steel  at  24 J; 
600  shares  Pressed  Steel  Car  at  34 ;  700  shares  Western  Union  Telegraph 
at  95^.  He  sold  the  Steel  stock  at  39^,  the  Pressed  Steel  Car  at  53|,  and 
the  Western  Union  at  92,  brokerage  being  \%  for  buying  and  for  selling. 
What  was  the  net  gain  or  loss  ? 

31.  A  certain  county,  on  Jan.  1,  1907,  issued  $250,000  worth  of  5% 
10-year  coupon  bonds.  If  these  bonds  were  sold  through  a  broker  at 
102|,  how  much  was  received  by  the  county?  Brokerage  |%.  If  the 
interest  is  payable  semi-annually,  what  is  the  amount  of  each  interest 
coupon?  How  much  must  be  levied  in  taxes  each  year  to  pay  the 
interest  and  provide  a  sinking  fund  sufficient  to  pay  the  bonds  in  full  at 
maturity?  What  would  be  the  annual  rate  to  be  levied,  if  the  assessed 
valuation  of  the  county  averages  $46,875,000? 

32.  A  broker  bought  for  a  customer  800  shares  of  United  States  Steel 
common,  at  a  total  cost  of  $30,100;  brokerage  \%.  Find  market  quota- 
tion of  stock. 

33.  How  much  must  be  paid,  including  brokerage  at  ^%,  for  a  suffi- 
cient number  of  United  States  4*s  at  123|  to  obtain  an  annual  income  of 
$1200? 

34.  My  broker,  after  selling  500  shares  of  Philadelphia  Gas  stock  and 
deducting  the  usual  commission,  remitted  $  534,312.50.  What  was  the 
market  quotation  ? 

35.  What  income  will  a  man  receive  from  an  investment  of  $6448  in 
United  States  coupon  4's  at  the  lowest  market  price  as  per  above  table  ? 


XX 

INSURANCE 

212.  Nature  of  Insurance.  Some  losses  or  damages  to 
property,  such  as  by  fire  or  tornado,  are  unavoidable.  Such 
losses  are  serious,  and  they  are  often  likely  to  bring  finan- 
cial ruin  to  owners.  To  avoid  such  calamities,  owners  of 
property  subject  to  losses  of  some  particular  kind  have  fre- 
quently banded  themselves  together  to  share  losses  among 
themselves.  By  agreeing  to  assist  in  making  good  the  loss 
to  whomsoever  it  might  fall,  they  secured  themselves  against 
disaster.  From  such  beginnings  has  grown  the  institution  of 
modern  insurance.  Insurance^  then,  consists  of  a  contract 
guaranteeing  to  make  good  a  loss  from  a  certain  cause. 
The  agreement  is  known  as  an  insurance  policy. 

213.  Kinds  of  Companies.  When  an  agreement  for  insur- 
ance is  made  between  those  mutually  interested,  to  mutually 
share  losses  from  a  particular  cause,  it  is  known  as  mutual 
insurance. 

When,  as  a  business  investment,  a  company  is  organized 
which  undertakes,  for  a  given  fee,  to  make  good  a  loss  during 
a  specified  time,  it  is  a  stock  insurance  company.  Sometimes 
a  stock  company  agrees  to  divide  all  earnings  of  the  company, 
above  a  specified  dividend  on  their  stock,  among  the  insured. 
It  thus  partakes  of  the  nature  of  both  stock  and  mutual 
companies. 

214.  Property  and  Personal  Insurance.  Whenever  it  is  a 
loss  in  property  which  is  insured  against,  it  is  property  insur- 
ance. Whenever  the  insurance  is  against  a  loss  due  to  sick- 
ness, accident  to,  or  death  of  a  person,  it  is  personal  insurance y, 

231 


232  ELEMENTS  OF  BUSINESS  ARITHMETIC 

PROPERTY  INSURANCE 

215.  Forms  of  Property  Insurance.  If  property  is  insured 
against  loss  from  fire,  it  is  fire  insurance;  if  against  loss  from 
wind  or  storm,  it  is  tornado  insurance;  if  against  loss  or 
damage  while  being  transported  by  land  or  by  sea,  it  is 
transit  insurance^  that  for  ship  or  cargo  lost  at  sea  being 
marine  insurance  ;  if  against  loss  or  damage  to  live  stock  by 
death,  disease,  lightning,  or  other  casualty,  it  is  live-stock 
insurance.  These  are  among  the  principal  forms  of  property 
insurance. 

216.  Kinds  of  Policies.  If  the  value  of  the  property 
insured  or  the  amount  of  the  indemnity  in  case  of  loss  is 
determined  and  agreed  upon,  at  the  time  the  policy  is  issued, 
it  is  known  as  a  closed  or  valued  policy.  If  the  real  value  of 
the  property  loss  is  open  for  determination  after  the  loss 
occurs,  regardless  of  the  face  of  the  policy  upon  which  the 
insurance  premium  has  been  paid,  it  is  an  open  policy. 
Many  states,  by  law,  have  declared  that  all  policies  must  be 
valued  policies.  This  requires  the  insurance  company  to 
pay  the  full  face  of  the  policy  in  case  of  a  total  loss,  regard- 
less of  the  actual  value  of  the  property. 

217.  Cost  of  Insurance.  In  mutual  companies,  the  cost  of 
insurance  depends  upon  the  amount  of  losses  suffered  by  the 
different  members  of  the  company.  This,  together  with  the 
actual  cost  of  carrying  on  the  business  of  the  company,  is 
apportioned  among  the  members,  in  the  form  of  assessments. 
In  stock  companies,  a  definite  fee  is  charged  for  insurance 
during  a  given  pieriod  of  time,  and  this  fee  is  called  a  premium. 
The  amount  of  the  premium  is  usually  a  per  cent  of  the  face 
of  the  policy,  or  the  amount  of  loss  which  it  is  agreed  to 
make  good.  This  is  known  as  the  rate  of  insurance.  It 
varies  with  the  kind  of  buildings,  their  location  with  refer- 
ence to  other  buildings,   efficiency  of   fire   protection,  etc. 


INSURANCE 


233 


Usually,  a  given  district  is  plotted,  and  the  rate  for  each 
building  is  fixed. 

Below  is  a  schedule  of  rates  for  such  a  plotted  district, 
expressed  as  a  certain  sum  for  each  $100  of  insurance. 
The  diagram  on  which  this  is  based  is  on  page  234. 


Table  of  Eates 


District  No. 


City  of 


KiSK 

Plot  No. 

Annual  Ratb 
PER  $  100 

Frame  Dwelling  and  Contents 

Brick  Store  and  Contents 

Brick  Church  and  Contents 

Brick  Business  Block  and  Contents     .     .     . 
Frame  Handle  Factory  and  Contents  .     .     . 
Brick  Livery  Barn 

1 

2 

3 

4 

5 

6 

•  7 

8 

9 

10 

11 

$.35 

.25 

.50 

.50 

1.75 

1.2j5 

Frame  Store  and  Dwelling,  and  Contents    . 

Frame  Barn  and  Contents 

Brick  Dwelling  and  Contents .     .     .     .     .     . 

Brick  Schoolhouse  and  Contents      .... 

Brick  Store  and  Dwelling,  and  Contents .     . 

.40 
1.00 
.18 
.50 
.30 

PROBLEMS 

1.  Find  the  cost  of  insuring  property  valued  at  $5000,  at  1^%. 

2.  At  2%>  how  much  insurance  can  I  procure  for  $148? 

3.  I  paid  $30  for  insuring  a  house  worth  $3200  at  |  its  valuation. 
What  was  the  rate  ? 

4.  Find  the  cost  of  insuring  each  of  the  buildings  in  the  table  above 
at  I  the  following  valuation  : 

Frame  Store  &  Dwelling        .  $2,450 

Frame  Barn  on  Above  Lot    .  500 

Brick  Dwelling 6,700 

Frame  Barn  on  Above  Lot     .  800 

Brick  Schoolhouse     ....  75,000 

Brick  Store  &  Dwelling      .     .  7,500 


Frame  Dwelling     .     .     . 

$2,400 

Brick  Store 

10,500 

Brick  Church     .... 

12,000 

Brick  Business  Block 

32,000 

Frame  Handle  Factory  . 

4,500 

Brick  Livery  Barn     .     . 

3,600 

234 


ELEMENTS  OF  BUSINESS  ARITHMETIC 


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133aiS 


133yiS 


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INSURANCE  235 

5.  The  contents  of  the  frame  dwelling  of  the  plot  are  valued  at 
$1100;  the  contents  of  the  brick  store,  at  $12,450;  the  contents  of  the 
brick  church,  at  $5500;  the  contents  of  the  brick  business  block,  at 
$25,600.  If  these  goods  are  insured  for  |  of  their  valuation,  what  would 
be  the  premium  on  each  lot  ? 

6.  An  agent's  premium  is  $25  for  insuring  a  house  at  |%.  What  is 
the  face  of  the  policy? 

7.  At  \%,  what  is  the  annual  premium,  at  |  valuation,  on  a  house 
worth  $16,000? 

8.  A  merchant  pays  $  150  for  insurance  on  his  stock  of  goods  at  f  %. 
What  is  the  amount  of  the  policy  ? 

9.  An  agent  receives  $112.50  for  insuring  a  house  for  80%  of  its 
valuation  at  |%.     What  is  the  value  of  the  house? 

10.  A  man  has  a  house  valued  at  $  24,000,  and  furniture  valued  at 
$6000.  He  insures  the  house  at  |  its  valuation,  and  the  furniture  at  | 
its  valuation.  What  is  the  annual  premium,  |%  for  the  house  and  f% 
for  the  furniture  ? 

11.  If  it  cost  $663  to  insure  a  building  valued  at  $132,600,  what  will 
it  cost  at  the  same  rate  to  insure  a  building  valued  at  $  105,000  ? 

12.  The  premium  on  a  house  valued  at  $10,500,  insured  at  f  its 
valuation,  is  $  47.25.     Find  the  rate  of  insurance. 

13.  If  a  house  and  its  contents  are  valued  at  $6500,  for  how  much 
must  it  be  insured,  at  1^%,  to  cover  loss  and  premium  in  case  of  total 
destruction  ? 

14.  A  cargo  of  coffee  valued  at  $35,000  was  insured  for  $20,000,  in  a 
policy  containing  an  average  clause.  It  was  damaged  to  the  amount  of 
$15,000.     How  much  should  the  company  pay? 

Note.  —  Under  an  "  average  clause  "  such  a  part  of  the  loss  is  paid,  as 
the  policy  is  of  the  real  value  of  the  property  insured. 

15.  A  steamer  is  insured  for  $75,000.  Its  value  is  $100,000.  If  it  is 
insured  at  2^%,  what  will  be  the  loss  to  the  company  in  case  of  total 
destruction  ? 

16.  An  agent  insures  a  cargo  of  cotton  costing  $  9775,  at  the  rate  of 
2\%,  for  an  amount  that  will  cover  the  cost  of  cotton  and  premium. 
What  is  the  face  of  the  policy  ? 

17.  A  speculator  bought  4000  bbl.  of  flour,  and  had  it  insured  for 
80%  of  its  cost,  at  3|%,-  He  paid  a  premium  of  $980.  At  what  price 
per  barrel  must  he  sell  it  in  order  to  gain  10  %  on  the  total  cost  ? 


236  ELEMENTS  OF  BUSINESS  ARITHMETIC 

18.  A  manufacturer  insured  his  factory  for  1 27,000,  and  its  contents 
for  $66,000.  He  paid  $700  for  premium  and  policy.  If  the  policy  cost 
$2.50,  what  was  the  rate  per  cent  premium? 

19.  A  dealer  in  New  York  ordered  his  Chicago  agent  to  buy  4000  bu. 
of  wheat  at  70^;  3000  bu.  at  30^;  7500  bu.  corn  at  37^  ^ ;  paying  2^% 
commission  for  buying.  The  grain  was  shipped  by  boat,  and  a  policy  at 
1-J%  was  taken  out  to  cover  the  cost  of  grain  and  commission.  What 
was  the  amount  of  the  policy,  and  what  the  amount  of  premium? 

20.  The  Atlas  Insurance  Company  insured  a  block  of  buildings  for 
$150,000,  at  75  }2^  per  $100.  Thinking  the  risk  too  great,  it  reinsured 
$  50,000  in  the  ^tna,  at  |  %  and  $  65,000  in  the  Manhattan,  at  |  %.  How 
much  premium  did  each  company  receive?  What  was  the  gain  or  loss 
to  the  Atlas?  What  per  cent  premium  did  it  receive  for  the  part  of  the 
risk  not  reinsured  ? 

21.  A  house  cost  $  8000 ;  it  was  insured  for  |  its  valuation  at  1^  %  for 
3  years.  What  would  be  my  loss  and  that  of  the  company,  if  the  house, 
were  totally  destroyed  by  fire  ? 

22.  A  residence  valued  at  $4-500  is  insured  for  |  its  value  at  |  %  per 
annum.  The  company  will  insure  the  house  for  3  years  on  a  payment 
of  2|  times  the  annual  premium  in  advance.  What  will  it  cost  to  insure 
the  house  for  3  years  ?  What  will  it  cost  to  insure  for  5  years,  if  the 
company  will  accept  4  annual  premiums  in  advance  as  payment  for  5 
years  ? 

23.  How  much  will  it  cost  to  insure  a  factory  valued  at  $75,000  at 
1%,  and  the  machinery  valued  at  $25,000  at  |%? 

PERSONAL  INSURANCE 

218.  Life  Insurance.  When  insurance  is  upon  the  hazard 
of  life,  it  is  life  insurance.  Upon  the  death  of  the  insured  it 
is  paid  to  an  heir  or  other  person  named  in  the  policy  as  the 
beneficiary. 

Life  insurance  may  be  paid  for  either  by  assessments  when 
deaths  occur,  by  a  stated  number  of  assessments  in  a  year,  or 
by  a  fixed  premium^  payable  monthly,  quarterly,  semi-annu- 
ally, or  annually.  When  a  fixed  premium  is  paid,  the  policy 
may  be  either  participating  or  nonparticipating  in  the  profits 
of  the  association.     The  premiums  may  be  payable  annually 


INSURANCE  237 

until  death  of  the  insured,  a  life  payment  policy  ;  or  may  be 
payable  annually  only  for  a  period  of  10,  15,  or  20  years,  a 
limited  payment  policy^  when  the  policy  is  said  to  be  "  fully 
paid  up."  The  latter  are  termed  10  payment  life,  20  pay- 
ment life,  etc. 

What  is  known  as  a  term  policy  may  also  be  purchased  for 
a  period  of  1,  5,  10,  15  or  20  years.  The  holder  is  insured 
for  the  term  only,  and,  if  desirous  of  continuing  the  insur- 
ance, must  purchase  another  policy  at  increased  rates,  because 
of  increased  age. 

When  a  fixed  premium  is  charged,  the  insurance  is  said  to 
be  "  old  line."  If  the  policy  is  a  participating  one,  the  profits 
of  the  company  may  operate  to  lessen  the  annual  premium, 
or  may  be  deferred  until  the  policy  matures  and  added  to 
the  face  value. 

Fraternal  organizations,  with  a  side  feature  of  insurance 
or  with  insurance  as  their  chief  purpose,  offer  one  method 
of  life  insurance,  often  with  additional  features  of  health  or 
accident  benefits.  They  usually  employ  the  assessment  plan, 
and  often  assess  for  more  than  enough  to  pay  the  death 
losses,  in  order  to  use  the  surplus  in  establishing  a  reserve 
fund  to  assist  in  paying  death  losses  when  the  membership 
becomes  older,  and  the  percentage  of  deaths  increase. 

219.  Endowment  Insurance.  Endowment  insurance  is  a 
modern  outgrowth  of  the  life  insurance  idea.  At  the  end  of 
the  stated  endowment  term,  e.g.  5,  10,  15,  or  20  years,  the  in- 
surance is  to  be  paid  to  the  insured  himself,  should  he  be 
alive  at  that  time.  Should  he  die  before  that  time,  the  full 
face  of  the  policy  is  to  be  paid  to  the  beneficiary  named  in 
the  policy.  While  retaining  the  life  insurance  idea,  it  thus 
adds  to  it  a  savings  or  investment  feature. 

220.  Accident  and  Health  Insurance.  Accident  or  casualty 
insurance  companies  pay  an  indemnity  when  one  is  injured 


238 


ELEMENTS  OF  BUSINESS  ARITHMETIC 


by  an  accident,  in  travel  or  otherwise.     The  amount  paid  is 
usually  graduated  to  the  extent  of  the  injury. 

Insurance  against  ill  health  may  also  be  had,  the  insured 
receiving  a  weekly  or  monthly  payment  while  sick. 

221.    Cost  of  Life  Insurance.     The  price  of  a  life  insurance 
policy  is  a  stated  annual  amount  or  premium  per  thousand 


Annual  Premium  Rates  for  $1000  Insurance 


WHOLE  LIFE 

ENDOWMENT  PREMIUM 

Aqb 

Life 

20  Yeaes 

15  Yeabs 

10  Years 

Age 

In  15  Yeabs 

In 
20  Yeaes 

21 

$19.47 

129.59 

§35.65 

$48.11 

21 

$67.03 

$49.07 

22 

19.91 

30.06 

36.20 

48.85 

22 

67.13 

49.17 

23 

20.36 

30.55 

36.78 

49.61 

23 

67.23 

49.27 

24 

20.84 

31.06 

37.38 

50.40 

24 

67.33 

49.39 

25 

21.34 

31.58 

38.00 

51.22 

25 

67.44 

49.52 

26 

21.86 

32.12 

38.63 

52.06 

26 

67.56 

49.65 

27 

22.41 

32.69 

39.63 

62.93 

27 

67.69 

49.79 

28 

22.99 

33.27 

39.98 

53.83 

28 

67  83 

49.95 

29 

23.59 

33.88 

40.70 

54.76 

29 

67.97 

60.11 

30 

24.23 

34.51 

41.43. 

65.73 

30 

68.12 

50.28 

31 

24.87 

35.14 

42.17 

56.70 

31 

68.29 

50.48 

32 

25.65 

35.81 

42.94 

57.71 

32 

68.46 

50.69 

33 

26.30 

36.52 

43.76 

68.78 

33 

68.66 

50.91 

34 

27.08 

37.26 

44.62 

59.88 

34 

68.87 

51.16 

35 

27.91 

38.04 

45.51 

61.03 

35 

69.09 

51.42 

36 

28.78 

38.85 

46.43 

62.21 

36 

69.34 

51.72 

37 

29.70 

39.70 

47.39 

63.44 

37 

69.60 

52.04 

88 

30.68 

40.59 

48.39 

64.71 

38 

69.90 

52.40 

39 

3L71 

41.51 

49.43 

66.03 

39 

70.21 

52.79 

40 

32.81 

42.49 

50.52 

67.40 

40 

70.56 

53.22 

41 

33.94 

43.50 

61.63 

68.80 

41 

70.95 

63.70 

42 

35.14 

44.55 

52.79 

70.24 

42 

71.37 

54.22 

43 

36.45 

45.69 

64.04 

71.77 

43 

71.85 

54.81 

44 

37.83 

46.90 

55.34 

73.36 

44 

72.37 

65.46 

45 

39.30 

48.17 

56.71 

75.02 

45 

72.96 

56.17 

INSURANCE  239 

dollars  of  insurance.    The  lowest  in  cost  is  in  purely  mutual 
companies,  like  fraternal  organizations. 

The  price  depends,  too,  upon  the  age  of  the  applicant  and 
upon  the  conditions  of  the  contract.  The  term  policy,  or  the 
straight  life-payment  nonparticipating  policy  may  be  pur- 
chased at  the  least  cost,  while  a  limited  payment  or  endow- 
ment policy  will  cost  proportionately  more.  Rates  vary,  but 
the  table  on  the  opposite  page  gives  a  fair  approximation  of 
the  cost  of  the  common  kinds  of  policies  in  standard  com- 
panies at  different  ages. 

PROBLEMS 
From  the  table  find  the  annual  premium  required  for : 

1.  A  life  policy  of  $3000,  age  30. 

2.  A  twenty-payment  life  policy  of  $  5000,  age  27. 

3.  A  twenty-year  endowment  policy  for  $4000,  age  32. 

4.  A  ten-payment  life  policy  for  $  3500,  age  35. 

5.  A  man  takes  out  a  twenty-payment  life  policy  for  $  3000  at  the 
age  of  25.  If  he  dies  at  the  age  of  40,  how  much  does  the  face  of  the 
policy  exceed  the  premiums  paid  ? 

6.  If  money  is  worth  6%,  what  do  the  premiums  in  problem  5 
amount  to?  How  much  does  the  face  exceed  that  amount?  (Annual 
interest.) 

7.  A  man  at  the  age  of  28  takes  out  a  straight  life  policy  and  a 
twenty-year  endowment  policy,  each  for  $  2000.  If  he  dies  at  40,  which 
gives  the  greater  returns  ? 

8.  A  man  30  years  of  age  took  out  an  endowment  policy  for  $  3000, 
payable  in  15  years,  and  died  after  making  six  payments.  How  much 
less  would  a  life  policy  have  cost  ? 

9.  A  man  aged  35  years  takes  out  an  endowment  policy  for  $  15,000, 
payable  to  himself  in  20  years,  or  to  his  heirs  if  he  dies  before  that  time. 
What  annual  premium  will  he  have  to  pay  ?  If  death  occurs  at  the  end 
of  the  ninth  year,  how  much  would  he  have  paid  out  in  premiums? 
How  much  less  would  a  twenty-payment  life  policy  have  cost  ? 

10.  At  the  age  of  32  a  man  takes  out  a  .|3500  life  policy,  and  at 
the  age  of  35  a  $  1000  twenty-year  endowment.  How  much  does  the 
insurance  exceed  the  premiums  paid,  if  he  dies  at  the  age  of  45? 


XXI 

PROPORTION 

222.  Ratio.  The  relative  size  of  two  numbers,  as  shown 
by  division  and  expressed  by  their  quotient,  is  called  their 
ratio.  Thus  the  ratio  of  8  to  4  is  2 ;  96  to  60  is  If ;  and  of 
25  to  50  is  J.  Instead  of  using  the  sign  of  division  (-?-)  to 
express  a  ratio,  the  horizontal  line  is  left  out,  making  the 
sign  simply  a  colon  (:),  e.g.  8 : 2  or  96  :  60.  The  meaning, 
however,  remains  the  same,  and  the  ratio  may  always  be 
found  by  dividing  the  first  number  or  term  by  the  second. 

223.  Comparison  of  Like  Quantities  Only.  Since  ratio 
shows  comparative  size,  it  may  only  exist  between  quantities 
like  in  kind.  Thus,  8  ft.  and  4  ft.,  or  8  houses  and  4  houses, 
may  be  compared,  while  8  boxes  and  4  horses  may  not.  8  ft. 
and  4  yd.  may  be  compared,  but,  to  do  so,  they  must  first  be 
reduced  either  to  feet  or  yards,  e.g.  the  ratio  of  8  ft.  to  12  ft., 
orf 

When  the  numbers  refer  to  units  of  measurement,  the 
quantities  being  measured  must  also  be  like  in  kind,  if  the 
ratio  of  the  quantities  being  measured  is  desired.  Thus, 
8  doz.  and  2  doz.  may  be  compared,  if  it  is  8  doz.  chairs  and 
2  doz.  chairs,  but  not  if  8  doz.  chairs  and  2  doz.  horses ;  or 
8  ft.  and  2  ft.  may  have  ratio  if  it  is  8  ft.  high  and  2  ft. 
high,  but  not  if  8  ft.  high  and  2  ft.  wide,  etc. 

224.  Order  of  Terms.  The  number  to  be  treated  as  a 
dividend  is  always  written  first.     The  ratio  of    6:3,  then, 

.would  be  2  and  not  |.     Since  the  number  of   which  it  is 

240 


PROPORTION  241 

desired  to  know  the  comparative  size  is  always  written  first, 
it  is  called  the  antecedent^  i.e.  the  one  which  goes  before. 
The  number  with  which  it  is  desired  to  compare  the  first  is 
always  written  second,  and  because  it  follows  the  other,  it  is 
called  the  consequent.  The  antecedent  and  its  consequent 
together  form  a  couplet. 

225.  Proportion.  The  ratio  of  two  different  couplets  may 
be  the  same.  Thus,  the  ratio  of  8  horses  to  2  horses  is  4, 
and  the  ratio  of  1 400  to  f  100  is  4.  Likewise,  the  ratio  of 
3  ft.  in  height  to  6  ft.  in  height  is  J,  and  the  ratio  of  3  days 
to  6  days  is  ^. 

Whenever  two  ratios  are  equal,  they  are  in  proportion. 
Thus,  as  the  ratio  of  8  horses  to  2  horses  is  4,  the  ratio  of 
their  cost  ($400  to  8100)  would  also  be  4.  We  would, 
therefore,  say  that  the  ratio  of  8  horses  to  2  horses  equals 
the  ratio  of  1400  to  $100. 

The  equality  of  related  ratios  is  usually  expressed  by  the 
double  colon,  e.g.  8  horses:  2  horses  ::  $400  :  $100.  This 
proportion  would  be  read  as  follows :  8  horses  is  to  2  horses 
as  $400  is  to  $100.  The  equality  sign  may  also  be  used, 
giving  the  above  proportion  the  form  of  8  horses  :  2  horses 
=  $400:  $100. 

226.  Means  and  Extremes.  As  there  must  be  two  ratios 
whose  equality  forms  a  proportion,  every  proportion  must 
have  four  terms.  When  written  formally  as  a  proportion, 
the  first  and  fourth  terms  are  called  the  extremes^  meaning 
the  outside  numbers.  The  second  and  third  terms  of  a  pro- 
portion are  called  the  means,  meaning  the  middle  numbers. 

Since  the  antecedents  of  each  ratio  are  the  dividends,  and 
the  consequents  are  divisors,  the  extremes  and  means  each 
consist  of  one  dividend  and  one  divisor.  The  quotient  of 
each  couplet  being  the  same,  the  product  of  the  dividend  of 
one  couplet  and  the  divisor  of  the  other  is  equal  to  the  prod- 


242  ELEMENTS  OF  BUSINESS  ARITHMETIC 

uct  of  the  other  dividend  and  divisor.  In  other  words,  the 
product  of  the  means  is  equal  to  the  product  of  the  extremes. 

The  above  being  true,  it  is  only  necessary  to  know  any 
three  of  the  terms  of  a  proportion  in  order  to  find  the  fourth. 
Thus,  in  the  proportion,  2  hats :  5  hats : :  13 :  fa;,  the  product 
of  the  means  (treating  the  terms  of  the  first  ratio  as  multi- 
pliers and,  therefore,  abstract  numbers)  is  f  15.  Since  this 
$15  is  the  product  also  of  the  extremes,  and  one  of  them  is 
2,  the  other  must  be  ^  of  $15,  or  $7.50.  The  completed  pro- 
portion, then,  would  be  2  hats :  5  hats : :  $3  :  $7.50. 

The  use  of  three  terms  to  find  the  fourth,  has  given  rise 
to  the  phrase  "  the  rule  of  three,"  which  is  a  name  formerly 
applied  to  solutions  by  proportion. 

227.  Directly  and  Inversely  Proportional.  Assuming  that 
each  man  does  an  average  day's  work,  the  larger  the  number 
of  men  employed  on  a  given  piece  of  work,  the  more  work 
is  done.  Thus,  if  2  men  can  wrap  and  pack  150  boxes  of 
oranges  in  a  day,  4  men  can  wrap  and  pack  300  boxes.  In 
other  words,  if  the  number  of  men  is  doubled,  twice  the  work  is 
accomplished.  Whenever  two  quantities  increase  or  decrease 
together  in  this  way,  and  with  a  constant  ratio,  they  are  said 
to  vary  directly  or  to  be  directly  proportional.  We  may  also 
say  that  the  amount  of  work  done  would  bear  a  direct  ratio 
to  the  number  of  men  employed. 

On  the  other  hand,  if  the  number  of  men  is  increased, 
the  same  amount  of  work  would  require  less  time  for  its 
completion.  Thus,  if  2  men  can  wrap  and  pack  600  boxes 
in  4  days,  4  men  could  do  it  in  half  the  time.  In  other  words, 
if  the  number  of  men  is  doubled,  the  time  required  is  but  one 
half  as  much.  Whenever  one  quantity  increases  as  another 
decreases,  or  decreases  as  another  increases,  keeping  the  ratio 
constant,  they  are  said  to  vary  inversely.,  or  to  be  inversely 
'proportional.     Thus,  the  time  required  to  do  a  given  piece 


PROPORTION  243 

of   work   bears    an    inverse    ratio   to   the    number    of   men 
employed. 

Query.  —  Which  of  the  following  are  directly  and  which 
inversely  proportional : 

1.  The  weight  of  coal  and  its  cost? 

2.  The  height  of  buildings  and  their  shadows? 

3.  The  number  of  workmen  and  the  amount  of  work 
done  in  a  given  time? 

4.  The  number  of  workmen  and  the  time  required  to  do 
a  given  amount  of  work? 

5.  The  speed  of  an  automobile  and  the  time  required  to 
travel  a  certain  distance? 

6.  The  weight  of  freight  and  the  freight  charges? 

7.  The  area  of  squares  and  the  length  of  their  sides? 

8.  The  amounts  loaned  and  interest  earned? 

9.  The  attraction  one  body  has  for  another  and  their  dis- 
tance apart? 

10.  The  greater  the  capital  stock  in  a  company  and  the 
size  of  the  dividend  which  can  be  declared  with  given  profits? 

11.  The  amount  of  assessable  property  and  the  tax  levies 
to  raise  a  given  amount? 

12.  The  amounts  loaned  and  the  rates  to  earn  the  same 
amount  of  interest? 

228.  Statement  and  Solution.  The  direct  proportion  given 
in  Section  227  would  be  stated  thus  : 

4  men :  2  men : :  300  boxes :  150  boxes. 

In  the  second  couplet,  300  boxes,  or  what  is  done  by  4 
men,  is  the  first  term,  just  as  4  men  is  the  first  term  of  the 
first  couplet ;  and  150  boxes  is,  likewise,  the  second  term 
of  the  second  couplet,  as  2  men  is  the  second  term  in  its 
couplet. 


244  ELEMENTS  OF  BUSINESS  ARITHMETIC 

The  inverse  proportion  would  be  stated  thus : 
4  men :  2  men : :  4  days :  2  days. 

This  order  is  necessary  that  the  ratio  of  each  couplet  be 
the  same  (in  this  case,  2).  It  will  be  noticed,  however,  in  the 
second  couplet,  that  2  days,  the  time  required  by  4  men,  is 
the  second  term  in  its  couplet,  although  the  4  men  is  the  first 
term  in  its  couplet.  Likewise,  the  4  days  is  the  first  term 
of  its  couplet,  although  the  two  men  working  is  the  second 
term.  In  other  words,  in  direct  proportion  the  order  of  the 
terms  in  the  second  couplet  is  the  same  as  in  the  first. 

In  inverse  proportion  the  order  of  the  terms  in  the  second 
couplet  is  the  inverse  of  the  order  in  the  first. 

Note.  —  As  a  matter  of  convenience  and  simplicity  in  solution,  it  will 
be  found  best  to  use  the  couplet  which  contains  the  unknown  or  required 
term  as  the  first  couplet  and  the  unknown  term  as  its  second  term. 

1.  2  men :  ?  men  : :  150  boxes :  300  boxes. 
300  boxes  x  2  =  600  boxes. 

600  boxes  -s- 150  boxes  =  4,  or  the  number  of  men  required. 

2.  2  men :  ?  men :  :  2  days :  4  days. 

4  days  x  2  =  8  days. 

8  days  -^  2  days  =  4,  the  number  of  men  required. 

In  general,  to  solve  a  proportion  when  it  is  correctly 
stated : 

1.  Divide  the  product  of  the  two  given  means  hy  the  one  given 
extreme^  or 

2.  Divide  the  product  of  the  two  given  extremes  hy  the  one 
given  mean. 

229.  Compound  Proportion.  Whenever  two  or  more  ratios 
are  equal  to  another  ratio,  the  proportion  is  said  to  be  com- 
pound. Compound  proportion,  therefore,  involves  three  or 
more  couplets,  all  having  the  same  ratio. 


PROPORTION  245 

The  statement  of  problems  containing  a  compound  pro- 
portion requires  the  comparing  of  every  couplet  to  one  basal 
couplet  (usually  taken  as  the  first  couplet),  ascertaining 
whether  the  proportion  be  direct  or  inverse,  and  arranging 
the  terms  of  the  second  couplets  accordingly. 

230.  Arranging  in  Couplets.  As  an  aid  to  comparing  the 
couplets  and  to  stating  the  proportions  involved,  it  will  be 
found  helpful  to  arrange  first  all  terms  given  in  the  problem, 
in  their  proper  couplets. 

Problem.  —  If  90  men  have  completed  the  construction 
of  3  miles  of  railroad  in  80  days,  how  many  men  should  be 
engaged  to  fulfill  a  contract  to  build  10  miles  of  road  in 
150  days  ? 

1.  Write  all  the  terms  belonging  together  in  a  column, 

thus, 

90  men 

3  miles 
80  days 

2.  On  the  other  side  of  a  vertical  line  drawn  to  the  right 
of  the  column,  write  the  remaining  terms,  arranging  like 
opposite  like,  forming  couplets,  thus, 


90  men 

? 

3  miles 

10 

80  days 

150 

231.  Statement  of  a  Compound  Proportion.  Using  the 
couplet  containing  the  unknown  term  as  the  first  couplet, 
the  following  couplets  should  be  compared  to  it  and  the 
terms  of  each  written  as  a  part  of  the  second  or  compound 
couplet,  their  order  depending  on  whether  the  statement  is 
of  direct  or  inverse  proportion.  Thus,  in  the  example  given, 
if  the  number  of  men  were  increased,  the  number  of  miles 
that  could  be  built  would  also  increase.     It  is,  therefore,  a 


246  ELEMENTS  OF  BUSINESS  ARITHMETIC 

direct  proportion,  and  the  order  in  both  couplets  would  be 
the  same,  e.g. 

90  men :  ?  men  :  :  3  miles  :  10  miles. 

But  if  the  number  of  men  were  increased,  it  would  take  a 
le%8  number  of  days  to  do  the  work.  This  proportion  would, 
therefore,  be  inverse,  and  the  second  couplet  would  be 
written  in  inverse  order,  thus, 

90  men ;  ?  men :  :  150  days  :  80  days. 

Combining  these  two  proportions  into  a  compound  propor- 
tion, we  have : 

Statement 

^^  o  3  miles :  10  miles 

90  men  :  ?  men  :  :  .,  r^  j  qa  j 

150  days :  80  days. 

232.   Solving  Compound  Proportion. 

10  miles  X  80  X  90  =  72,000  miles. 
3  miles  x  150  =  450  miles. 
72,000  miles -f- 450  miles  =  160,  or  the  number  of  men  required. 

Solution  by  Cancellation 

2 

AA  o  ?  miles :  10  miles 

^  men  :  ?  men  :  :  -•L   -,  qV,  , 

^^  }.p^  days  :  80  days. 

2  men  x  80  =  160  men.     An%. 

Note.  —  All  factors  removed  by  cancellation  must  be  taken  out  of 
hoik  the  means  and  the  extremes. 

PROBLEMS 

1.  What  effect  does  the  multiplying  or  dividing  of  both  terms  of  a 
ratio  have  on  its  value  ? 

2.  What  effect  does  multiplying  or  dividing  the  antecedent  have  on 
the  ratio  ?    Multiplying  or  dividing  the  consequent? 


PROPORTION  247 

What  is  the  ratio  of : 

3.  $8  to  $2?    $|to|i?    $.50  to  $.12^?    $26to|5.20? 

4.  100  to  25?     100tol4f?    33itol00?     100  to  6f  ? 

5.  $  .50  to  $  .15 ?    2  m.  to  40  cm.  ?     15  hr.  to  a  day? 

6.  What  number  has  to  40  the  ratio  of  2  ?    Of  ^?     To  5  the  ratio  of 
5?    To  15  the  ratio  of  I  ?     Of  3  ?     To  84  of  7?     Of  }'} 

7.  28  has  the  ratio  of  2  to  what  number  ?     12^  has  the  ratio  of  |  to 
what  number?     |  has  the  ratio  of  3  to  what  number? 

8.  A  pound  of  coffee  costs  40^,  and  of  butter  25  j^.     What  was  the 
ratio  of  their  costs  ? 

9.  The  diameter  of  a  circle  is  7  ft.  and  the  circumference  is  22  ft. 
What  is  the  ratio  of  the  circumference  to  the  diameter? 

10.  If  a  map  is  drawn  on  a  scale  of  1  in.  to  1  mi.,  in  what  ratio  are 
the  dimensions  diminished? 

11.  One  door  is  6  ft.  6  in.  by  3  ft.  8  in. ;  another  is  7  ft.  6  in.  by  4  ft. 
What  is  the  ratio  of  the  first  to  the  second?    Of  the  second  to  the  first? 

Examine  each  of  the  problems  from  12  to  30,  and  tell  whether  the 
ratio  is  direct  or  inverse.     Solve. 

12.  If  6  horses  cost  $  1200,  what  will  10  horses  cost  at  the  same  rate  ? 

13.  If  12  yd.  of  cloth  cost  $ 20,  what  will  35  yd.  cost? 

14.  If  6  horses  cost  $  300,  how  many  can  be  bought  for  $900? 

15.  If  15  yd.  of  silk  cost  1 22.50,  how  many  yards  can  be  bought 
for  $36? 

16.  If  it  takes  48  yd.  of  carpet  1  yd.  wide  to  cover  a  floor,  how 
many  yards  will  it  take  of  carpet  |  yd.  wide  ? 

17.  $200  earns  $  12  interest.     How  much  interest  will  $350  earn? 

18.  A  merchant  pays  $  6  freight  on  1000  lb.  of  merchandise.  What 
rate  is  that  per  100  lb.  ? 

19.  When  coal  is  worth  $9  a  ton,  what  will  1200  lb.  cost? 

20.  A  man  with  an  income  of  $  1000  saved  $  300.  The  next  year  his 
income  was  $  1200  and  he  saved  a  proportional  amount.  How  much  did 
he  save  ? 

21.  At  a  certain  time  of  day,  a  post  4  ft.  high  casts  a  shadow  3  ft. 
long.     What  is  the  height  of  a  tree  that  casts  a  shadow  of  15  ft.  ? 


248  ELEMENTS  OF  BUSINESS  ARITHMETIC 

22.  A  pipe  discharging  6  gal.  a  minute  can  fill  a  cistern  in  4  hr. 
How  long  will  it  take  a  pipe  discharging  8  gal.  a  minute  to  empty  it? 

23.  If  a  man  sells  f  of  his  farm  for  $4200,  what  would  |  of  it  be 
worth  at  the  same  rate  ? 

24.  A  hall  was  paved  with  tiles  9  in.  square,  and  640  were  used. 
How  many  tiles  6  in.  square  would  it  take  ? 

25.  If  a  tower  40  ft.  high  casts  a  shadow  70  ft.  long,  how  long  a 
shadow  will  a  tower  110  ft.  high  cast  ? 

26.  If  it  cost  1 60  to  make  a  walk  10  ft.  wide  and  180  ft.  long,  how 
much  will  it  cost  to  make  a  walk  8  ft.  wide  and  450  ft.  long  ? 

27.  If  3000  bricks,  each  8  in.  long  and  4  in.  wide,  will  lay  a  walk,  how 
many  bricks  6  in.  square  would  it  take  ? 

28.  If  it  cost  $  168  to  roof  a  space  72  ft.  long  and  21  ft.  wide,  how 
much  will  it  cost  to  roof  a  space  66  ft.  long  and  27  ft.  wide  ? 

29.  If  170  bu.  of  oats  feed  120  horses  34  days,  how  long  would  150  bu. 
feed  90  horses  ? 

30.  If  12  men,  in  4  da.  of  8  hr.  each,  earn  $  152.60,  at  the  same  rate, 
how  much  will  22  men  earn  in  5  da.  of  9  hr.  each  ? 

31.  10  men  can  pave  a  street  30  ft.  long  and  48  ft.  wide  in  2  da. 
How  many  men  will  it  take  to  pave  a  street  400  ft.  long  and  36  ft.  wide 
inl2|da.? 

32.  If  $290.70  interest  accrues  on  $1020  at  6%  for  4  yr.  9  mo.,  how 
much  interest  must  be  paid  on  $2700  at  7^%  for  3  yr.  4  mo.  ? 

33.  If  $  675,  put  at  interest  at  8  %,  earns  $  9  interest  in  60  da.,  in  how 
many  days  will  $1240  earn  $28.80  interest  at  6%  ? 

34.  If  20  men  working  12  da.  of  8  hr.  each  can  cut  400  cd.  of  wood, 
how  many  cords  should  12  men  cut  in  15  da.  of  10  hr.  each? 

35.  If  a  piece  of  timber  11  ft.  long,  10  in.  wide,  and  8  in.  thick  weighs 
1848  lb.,  find  the  length  of  another  piece  of  timber  which  weighs 
6048  lb.,  and  which  is  6  by  24  in.? 

36.  If  10  horses  eat  16  bu.  16  qt.  oats  in  9  da.,  how  many  days,  at  the 
same  rate,  will  123  bu.  28  qt.  feed  34  horses? 

37.  If  21  men  can  build  a  wall  28  rd.  long  in  96  da.,  how  many  men 
will  be  required  to  build  31^  rd.  in  84  da.  ? 


/ 


PROPORTION  249 

38.  If  12  compositors  in  60  da.  of  10  hr.  each  set  up  50  sheets  of  16 
pages  each,  32  lines  on  a  page,  in  how  many  days  of  8  hr.  can  18  com- 
positors set  up,  in  the  same  type,  128  sheets  of  12  pages  each,  40  lines  to 
the  page  ? 

39.  A  contractor  engaged  to  lay  20  mi.  of  road  in  300  da.  At  the  end 
of  80  da.  he  finds  that  90  men  have  laid  3  mi.  How  many  more  men 
must  he  engage  to  finish  the  work  in  the  required  time  ? 

40.  If  54  T.  of  anthracite  coal  can  be  stored  in  a  bin  28  x  20  x  4  ft., 
how  many  tons  can  be  stored  in  a  bin  45  x  18  x  8  ft.  ? 

41.  If  a  mow  10  x  6  x  8  yd.  holds  32  T.  of  hay,  how  deep  must  a 
mow  be  that  is  24  ft.  long  and  15  ft.  wide,  in  order  to  hold  86  T.  ? 

42.  If  60  men  make  an  embankment  f  of  a  mile  long,  30  yd.  wide, 
and  7  yd.  high  in  42  da.,  how  many  men  will  it  take  to  make  an  em- 
bankment 1000  by  36  yd.  and  22  ft.  high  in  30  da.  ? 

43.  If  50  men  can  do  a  piece  of  work  in  48  da.  working  8  hr.  a  day, 
how  many  hours  a  day  would  50  men  have  to  work  in  order  to  do  the 
same  work  in  32  da.  ? 

44.  If  the  interest  on  $84  at  6%  for  3  yr.  is  |15.12,  what  sum  must 
be  loaned  at  8%  for  1  yr.  6  mo.,  to  earn  the  same  amount? 

45.  If  I  loan  $600  for  8  mo.  and  get  $20  interest,  for  what  time  must 
I  loan  $1200  at  the  same  rate  to  get  $  90  interest? 


XXII 

PROPORTIONAL  PARTS  AND  PARTNERSHIP 

233.  Partnership.  The  association  of  two  or  more  persons 
in  a  business  firm,  or  partnership^  has  already  been  outlined 
under  Stocks  and  Bonds  (Sec.  196).  During  the  progress  of 
a  business,  it  often  becomes  necessary  to  take  an  inventory  of 
the  financial  condition  of  the  firm.  For  this  purpose,  a  state- 
ment of  resources  and  liabilities  is  made.  Under  resources 
are  listed  all  property  on  hand  and  all  accounts  owed  to  the 
business,  and  under  liabilities  all  debts  of  the  firm.  This 
casting  up  of  accounts  will  show  whether  the  profits  of  the 
business  exceed  the  expenses  (net  gain^  or  whether  the  ex- 
penses have  been  greater  than  the  profits  (net  loss}.  It  will 
also  show  whether  the  firm  has  sufficient  resources  to  meet 
all  liabilities,  in  which  case  it  is  solvent,  or  if  its  liabilities 
are  greater  than  its  resources,  when  it  is  insolvent.  From 
these  statements,  too,  may  be  computed  the  present  financial 
condition  of  the  firm  (the  net  resources  after  allowing  for 
all  liabilities),  which  is  termed  the  present  worth. 

234.  An  Application  of  Proportion.  The  distribution  of 
the  profits  or  apportionment  of  the  losses  of  a  business  part- 
nership is  often  so  simple  as  to  be  easily  resolved  into  frac- 
tional parts.  The  conditions  of  partnership  may,  however, 
become  quite  complicated,  involving  different  forms  and 
amounts  of  investments,  withdrawals,  and  increases,  different 
lengths  of  time,  etc.  In  such  cases  the  apportionment  of 
profits  or  losses  are  often  easily  and  quickly  accomplished 
by  proportion. 

250 


PROPORTIONAL  PARTS  AND  PARTNERSHIP      251 

235.  Partitive  Proportion.  The  form  of  proportion  used 
for  that  purpose  is  known  as  partitive  proportion  or  propor- 
tional pojrts.  The  term  "  partitive  proportion  "  means  parti- 
tioning a  whole  into  parts,  proportionally.  The  parts  of  the 
profits  belonging  to  each  partner  would  bear  the  same  ratio, 
i.e,  be  proportional  to  the  investments  made,  the  time  the 
capital  was  used,  or  some  other  definite  ratio  that  may  be 
ascertained. 

Thus,  if  A  invested  1500,  and  B  invested  flOOO,  a  profit 
of  1750  would  be  divided  into  8  250  and  |500,  respectively. 
The  sum  of  all  the  parts  invested  would  be  $1500,  and  A 
would  be  entitled  to  ^^%  (or  J)  of  all  the  profits,  which 
would  be  8250. 

Stated  proportionally,  A's  profit  would  be  found  by  solv- 
ing the  proportion 

1500: 11500::?:  $750 
1375,000  H- 11500  =  250,  or  1 250  profit. 
B's  profit  would  be  : 

11000:11500::?:  $750 

ig^  =  500,  or  $  500  profit. 

236.  Equivalent  Investments.  When  investments  in  the 
business  are  made  for  different  lengths  of  time,  the  profits 
or  losses  are  often  distributed  in  proportion  to  the  equiva- 
lent investments.  By  equivalent  investment  is  meant  the  sum 
which,  invested  for  a  unit  of  time,  is  equivalent  to  various 
sums  invested  for  different  periods  of  time. 

Thus,  if  A's  $500  were  invested  for  6  mo.  and  B's  $1000 
for  4  mo.,  the  profit  of  $750  would  not  be  distributed  as  \ 
and  |.  A's  $500,  invested  for  6  mo.,  would  be  equivalent 
to  $3000,  invested  for  1  mo.  ;  and  B's  to  $4000,  invested 
for  1  mo.  Their  total  equivalent  investments,  then,  would 
be  $7000,  and  A's  profits  would  be  f  of  $750,  or  $321.43. 


252  ELEMENTS  OF  BUSINESS  ARITHMETIC 

Proportionally  solved,  A's  profits  would  be 

13000  :|T000::?:  1750 

12,250,000  ^  $7000  =  321.43.  or  1321.43  profit. 

When  a  partner's  capital  is  increased  or  decreased  during 
the  term  he  remains  a  partner,  his  equivalent  investment  is 
found  by  finding  equivalent  investments  for  a  unit  of  time 
for   each    different    capital,   and    adding    such    equivalent" 
investments. 

Thus,  if  $1000  was  invested,  and  increased  1500  after 
4  mo.,  and  again  increased  $500  after  another  2  mo.,  and 
decreased  $800,  6  mo.  later,  where  it  remained  for  10  mo. 
longer  before  there  was  a  distribution  of  profits,  equivalent 
investments  would  be  found  in  the  following  way : 

$1000  X    4  =   $4,000,  equivalent  investment  for  1  mo. 

1500  X    2  =      3,000,  equivalent  investment  for  1  mo. 

2000  X    6  =    12,000,  equivalent  investment  for  1  mo. 

1200  X  10  =    12,000,  equivalent  investment  for  1  mo. 

$31,000  equivalent  investment  for  1  mo. 

237.  Adjustments  by  Interest.  Inequalities  in  amounts 
and  time  of  investments,  especially  when  there  are  increases 
and  withdrawals  of  investments  at  irregular  periods,  are 
often  adjusted  b}^  allowing  interest  on  all  investments  and 
charging  interest  on  all  sums  withdrawn.  The  profits  or 
losses  remaining  after  interest  has  been  allowed  or  charged, 
may  then  be  divided  equally  or  according  to  any  fixed  ratio. 

Thus  at  6  %  A's  $500  would  bear  $15  interest  in  6  mo.  If 
$200  were  withdrawn  at  that  time  for  the  remaining  6  mo., 
his  $300  would  earn  $9  in  the  remaining  6  mo.,  and  he  would 
be  charged  $6  interest  on  such  withdrawal.  His  net  in- 
terest earned  would,  therefore,  be  $15  +  $9  -  $6  =  $18. 
B's  $1000  for  4  mo.  would  be  allowed  $20  interest,  and 
$400  for  8  mo.,  $16,  and  if  $600  were  withdrawn,  he  would 


PROPORTIONAL  PARTS  AND  PARTNERSHIP      253 

be  charged  $24  for  the  remaining  8  mo.,  leaving  a  net  in- 
terest earning  of  $12.  A's  earned  interest  of  $18  and  B's 
of  $12  must  first  be  paid  out  of  the  $750  profit,  leaving 
$720  to  be  divided  in  proportion  to  the  original  investment. 
Of  this,  A  would  get  J,  or  $240,  plus  his  interest  of  $18,  or 
$258,  and  B  would  get  |,  or  $480,  plus  his  interest  of  $12, 
or  $492. 

Note.  —  When  interest  is  allowed  and  charged  on  capital  increases 
and  withdrawals,  net  profits  ate  often  shared  equally,  after  interest  has 
been  paid. 

PROBLEMS 

1.  Divide  360  into  parts  proportional  to  3  and  6. 

2.  Divide  $800  into  parts  proportional  to  1,  3,  and  6. 

3.  Divide  $240  into  parts  proportional  to  ^  and  ^. 

4.  Divide  $780  among  three  persons,  whose  shares  will  be  in  pro- 
portion to  ^,  ^,  and  ^. 

5.  A,  B,  and  C  engage  in  business  for  1  yr.  A  puts  in  $5000,  B 
$3000,  and  C  $2000.     If  they  gain  $3600,  what  is  each  man's  share? 

6.  Divide  $2400  among  A,  B,  and  C,  so  that  A's  part  will  be  twice 
C's  and  ^  B's. 

7.  The  total  receipts  of  a  gold  mining  company  for  1  yr.  were 
$15,750,000.  The  expenses  were  to  the  net  earnings  as  12  to  3.  What 
were  the  expenses  ?    The  net  earnings  ? 

8.  Divide  the  simple  interest  on  $65,000  for  1  yr.  8  mo.,  at  5|%, 
among  A,  B,  and  C,  so  that  A's  part  is  8  times  C's  and  |  B's. 

9.  A,  B,  and  C  pay  $75.60  for  a  pasture.  A  puts  in  10  horses,  B 
24  cows,  and  C  120  sheep.  If  3  sheep  eat  as  much  as  1  cow  and  2  cows 
as  much  as  3  horses,  what  rent  must  each  pay? 

10.  The  annual  earnings  of  a  steamship  company  were  $39,000,000. 
Find  the  amounts  received  from  freight  charges,  from  passenger  service, 
and  from  other  sources,  if  they  were  in  the  proportion  of  7  : 4  :  2. 

11.  The  holdings  of  the  shareholders  of  a  corporation  are  32,  13,  22, 
20,  50,  34,  42,  and  72  shares,  respectively.  If  a  dividend  of  $12,825  is 
divided  among  them,  what  does  each  shareholder  receive  ? 


254  ELEMENTS  OF  BUSINESS  ARITHMETIC 

12.  C  and  D  engaged  in  business  and  gained  $3500.  C's  capital 
was  $8000,  and  D's  was  $6000.  Find  each  partner's  share,  if  the  profits 
were  divided  according  to  investment. 

13.  Two  men  owned  a  carriage  factory.  One  had  invested  $75,000, 
the  other  $45,000.  The  net  earnings  for  1  yr.  were  $12,200.  What 
was  each  partner's  share  ? 

14.  A,  B,  and  C  enter  into  partnership  with  a  joint  capital  of 
$130,000  ;  A  furnishes  i,  B  I,  and  C  the  remainder.  Their  net  gain  is 
30%  of  the  amount  invested.     Find  each  man's  share  of  the  gain. 

15.  Dunn  and  McDonald  formed  a  partnership  in  which  Dunn  in- 
vested $5000  and  McDonald  invested  $2500.  The  gains  were:  mer- 
chandise, $940.25;  real  estate,  $356.50;  losses,  expense,  $420.  What 
was  the  net  gain  ?  What  was  the  gain  of  each  partner  ?  What  was  each 
partner's  present  worth  at  the  close  of  the  business  ? 

16.  A  and  B  formed  a  partnership  with  a  capital  of  $10,000.  A 
furnishes  $4000  and  B  $6000.  After  16  mo.  A  withdrew  $500,  and  at 
the  end  of  18  mo.  B  withdrew  $1000.  At  the  end  of  2  yr.  the  partner- 
ship was  dissolved  and  a  profit  of  $8750  was  divided.  How  much  did 
each  partner  receive  ? 

17.  A  and  B  engaged  in  a  dry  goods  business  for  3  yr.  from  April  1, 
1907.  Each  invested  $1800.  July  1,  1907,  A  increased  his  investment 
$350,  and  B  withdrew  $300;  Feb.  1,  1908,  each  withdrew  $800;  Feb.  1, 
1909,  each  invested  $1200.  There  was  a  gain  of  $1800  on  March  1. 
How  should  it  be  divided  ? 

18.  C  and  D  formed  a  partnership,  C  investing  $9000,  and  B  $12,000. 
It  was  agreed  that  B  should  take  $3000  from  the  gains  before  a  division 
was  made,  and  that  the  net  gain  or  loss  should  then  be  shared  equally. 
The  gains  were  $7580  and  the  losses  $1275.  What  was  the  net  gain  of 
each  partner?     The  present  worth  at  dissolution? 


INDEX 


Reference  to  Pages 


Acceptance,  212. 

Accident  insurance,  327. 

Accounts,  46  ;  savings  bank,  206. 

Account  sales,  180. 

Addition,  1 ;  by  groups,  2 ;  two-col- 
umn, 2;  of  decimals;  25;  of  frac- 
tions, 56. 

Ad  valorem  duties,  188. 

Angles,  98. 

Annual  interest,  194. 

Antecedent  in  ratio,  241. 

Apothecaries',  liquid  measure,  121 ; 
weight,  137. 

Arabic  notation,  23. 

Area,  72 ;  of  rectangles,  73 ;  non- 
rectangles,  101-104 ;  quadrilaterals, 
101 ;  trapezoids,  102  ;  triangles,  102- 
104 ;  circles,  105 ;  other  surfaces, 
107;  metric,  151. 

Articles   of   incorporation,    222. 

Assessed  valuation,  185. 

Assessments,  on  stocks,  222;  insur- 
ance, 232. 

Authorized  capital,  222. 

Averaging,  34. 

Avoirdupois  weight,  138. 

Bank  discount,  213. 

Banks,  205 ;    national,  205 ;    state  and 

private,  205  ;  savings,  206. 
Base  lines,  94. 

Bears,  in  stocks  and  bonds,  225. 
Beneficiary  of  insurance,  236. 
Bills,  46  ;  of  foreign  exchange,  145. 
Board  measure,  115. 
Bonds,  193,  223. 
Bricklaying,  114. 
Brokerage  and  brokers,  225. 
Bulk,  measures  of,  119. 


Bulls,  in  stocks  and  bonds,  225. 
Bushel,  unit,   119;    weights,   138. 

Calendar  months,  126. 

Capacity,  in  dry  units,  120 ;  liquids, 
121. 

Capital  stock,  222. 

Carpeting,  86. 

Cashier's  checks,  210. 

Casting  out  nines,  4. 

Certificates  of  deposit,  210 ;  of  stock, 
222. 

Certified  checks,  209. 

Change,  making,  10 ;  memorandum,  50. 

Chattel  mortgage,  192. 

Check  on  addition,  4. 

Checks,  209. 

Cipher,  for  marking  goods,  177. 

Circle,  101 ;  area  of,  105 ;  measure- 
ment of,  128. 

Circumference,  101 ;  ratio  to  diam- 
eter, 105. 

Civil  service,  addition  method,  4. 

Closed  insurance  policy,  232. 

Coins,  authorized,  141. 

Collection  by  draft,  211. 

Combinations,  addition,  1. 

Commercial  month,  127. 

Commission,  180,  225 ;  on  purchases, 
181. 

Common  denominator,  57 ;  divisor,  34 ; 
stock,  222. 

Compound  interest,  193 ;  proportion, 
244. 

Cone,  123 ;  surface  of,  107 ;  volume, 
124. 

Consignment,  46,  180. 

Cord,  of  wood,  113  ;  of  stone,  114. 

Corporations,  221. 


265 


256 


INDEX 


Correction  lines,  95. 

Cost  of  articles,  41 ;  per  hundred, 
43 ;    per  thousand,  43 ;    per  ton,  45. 

Couplet,  in  proportion,  241,  245. 

Coupons,  193,  224. 

Cube  root,  112. 

Cubes,  table  of,  113. 

Cubic  units,  109. 

Cylinder,  122 ;  surface  of,  107 ;  vol- 
ume, 124. 

Date   line,   international,    132. 

Days  of  grace,  195! 

Decimal  equivalents  ot  common  frac- 
tions, 40. 

Decimal  fractions,  62. 

Decimal  notation,  20  ;  digit  value  in,  5. 

Decimal    point,    use    of,    23. 

Decimals,  23 ;    square  root  of,  80. 

Degree,  128. 

Demand  certificates,  210. 

Denominator,  53. 

Direct  ratio  and  proportion,  242. 

Discount,  trade,  172 ;  series  in,  173 ; 
bank,  213 ;  true,  215 ;  fractional, 
173 ;  is  interest,  215 ;  selling  at, 
223,  224. 

Dividends,  222. 

Divisibility,  tests  of,  33. 

Division,  two  kinds  of,  18 ;  long  divi- 
sion, 20 ;  shortening  of,  20 ;  of  deci- 
mals, 26 ;  by  1  with  ciphers,  26 ; 
of  fractions,   56,   61. 

Divisors,  common,  34. 

Drafts,  143,  210. 

Drill  cards,  1,  17. 

Drill  tables,  addition,  1,  2 ;  multi- 
plication, 16. 

Dry  measure,  119. 

Duties,  188. 

Endowment  insurance,  237. 

Equation   in  explanations,  13,  17,  18. 

Equivalent  investments,  251. 

Even  numbers,  33. 

Exact  interest,  195. 

Excises,  189. 

Exchange,      142;     bank,      143,     210; 

by  wire,  144  ;  foreign,  144  ;  by  cable, 

147. 
Explanations,  suggestions   for,   13,   17. 
Extremes  in  proportion,  241. 


Factors,  33. 

Fire  insurance,  232. 

Firms,  221. 

Flooring,  85. 

Foreign  exchange,  144. 

Fractional  parts,  32 ;    in  division,  18 ; 

finding,  32,  36;    of  a  dollar,  41. 
Fractions,     53 ;      and     decimals,     62, 

square  root  of,  80. 

Gain,  157,  250. 
Geographical  mile,  129. 
Government  bonds,  224. 
Gram,  152. 

Greatest  common  divisor,  34. 
Groups,  adding  by,  2. 

Indorsement,  212. 

Indorsers,  192. 

Insolvency,  250. 

Inspection,  reduction  by,  55. 

Insurance,  231. 

Integers,  23. 

Interest,  192;  sixty-day  method,  196; 
6%,  199;  at  any  rate,  197;  tables 
of,  199 ;  days,  in  savings  banks, 
206 ;    in    partnership,    252. 

International  date  line,  132. 

Inverse  ratio  and  proportion,  242. 

Investments,  251. 

Invoicing,  46. 

Isosceles  triangle,  99 ;  area  of,  103. 

Kilogram,  152. 

Land  measure  and  survey,  94. 

Lathing,  88. 

Latitude,  128. 

Least  common  multiple,  34,  57. 

Legal,    limitations    to    interest,     194 ; 

rate  of  interest,  194 ;   tender,  142. 
Length,  measures  of,  67 ;    metric,  150. 
Letters  of  credit,   145. 
Liabilities  in  partnership,   250. 
Liability  of  stockholders,  223. 
Life  insurance,  236 ;    policy,  237 ;    cost 

of,  238. 
Limited  payment  policy,  237. 
Linear  measure,  67. 
Liquid    measure,    120;    apothecaries', 

121. 
Listing  goods,  177. 


INDEX 


257 


List  price,  172. 

Liter,  152. 

Live-stock  insurance,  232. 

Log  measure,  116. 

Longitude  and  time,  128;  diEference 
in,  129 ;  reduction  in,  131 ;  of  lead- 
ing cities,  134. 

Long  ton,  138. 

Loss,  157,  250. 

Lumberman's  reference  table,  118. 

Lumber  measure,  115. 

Maker,  of  note,  193;  draft,  209. 

Manifest,  or  customs  declaration,   188. 

Marine  insurance,  232. 

Marked  price,  172. 

Market  quotations,  225. 

Marking  goods,  177. 

Maturity,  192  ;  value,  213. 

Means  in  proportion,  241. 

Merchant's  rule,  199. 

Meter,  149. 

Metric  system,  149. 

Mixed  number,  53. 

Money  order,  P.O.,  143 ;  express,  143  ; 
foreign,  146. 

Mortgage,  192. 

Multiples,  33 ;    common,  34. 

Multiplication,  17;  drill  table,  17; 
long,  19 ;  of  decimals,  26  ;  by  1  with 
ciphers,  26 ;    of  fractions,  58. 

Mutual  insurance,  231. 

National  banks,  205. 

Nautical  units,  68. 

Negotiable  paper,  209. 

Non-assessable  stock,  223. 

Notes,  192;  negotiable  and  non-,  210. 

Numerator,  53. 

Odd  numbers,  33. 

Open  insurance  policy,  232. 

Papering,  92. 

Paper  money,  authorized,   141. 

Parallel,  99. 

Parallelograms,  100 ;   area   of   oblique, 

101. 
Parallels  of  latitude,  128. 
Partial  payments,  199. 
Partitive  proportion,  251. 


Partnership,  221 ;  by  proportion,  250. 

Par  value,  222. 

Payee,  of  notes,  193 ;   of  drafts,  209. 

Pay  rolls,  49. 

Per  capita  tax,  186. 

Percentage,  157;  finding  50%,  25%, 
20%).  158;  33i%,  163%,  12J%, 
14?%,,  160;  10%,  1%,  5%,  i%, 
162 ;    finding  other  per  cents,  164. 

Periodic  interest,  194. 

Personal  property,  185 ;   tax,  185. 

Pi  (TT),  105. 

Plastering,  88. 

Pointing  off,  in  decimals,  24 ;  in  mul- 
tiplication, 27  ;    in  division,  28. 

Policy,  of  insurance,  231 ;  partici- 
pating, 236. 

Poll  tax,  186. 

Polygons,  99  ;   area  of,  101. 

Ports  of  entry,  188. 

Preferred  stock,  222. 

Premimn,  selling  at,  223 ;  on  insurance, 
232. 

Present  worth,  215,  250. 

Prime  numbers,  33. 

Principal,   199 ;    meridians,  94. 

Prisms,  122;  volume  of ,  110,  123;  sur- 
face, 107. 

Proceeds,  215. 

Profit  and  loss,  157. 

Promissory  notes,  192. 

Proof,  in  addition,  4. 

Proper  fractions,  53. 

Proportion,  240;    in  partnership,  250. 

Proportional  parts,  251. 

Pyramids,  surface  of,  107;  volume, 
123. 

Qtcadrilaterals,  100;    area  of,  101. 
Quotient,  denomination  of  partial,  20. 

Rate,  in  interest,  192,  194 ;  insurance, 
232. 

Ratio,  240. 

Reading  problems,  19  ;  of  decimals,  23. 

Real  estate,  185. 

Reduction,  of  fractions,  54 ;  by  in- 
spection, 55. 

Registered  bonds,  224. 

Remainder,  exact  in  division,  30. 

Reserve  fund,  insurance,  237. 

Resources,  250. 


258 


INDEX 


Right  angle,  99. 

Roofs  and  roofing,  82. 

Roots,     square,     76;      application    to 

right  triangle,  80 ;    of  decimals  and 

fractions,  80  ;  cube,  112. 
Rule  of  three,  242. 

Scalene  triangles,  99  ;  area  of,  104. 

Sections  of  land,  96. 

Short,  selling,  225. 

Simplified  processes,  in  division  and 
multiplication,  37. 

Sinking  fund,  224. 

Six  per  cent  method,  199. 

Sixty-day  method,  196. 

Smuggling,  189. 

Solar  day  and  year,  126. 

Solvency,  250. 

Specific  duties,  188. 

Sphere,  123  ;  surface  of,  107 ;  volume, 
124. 

Square,  75,  100 ;  root,  76. 

Standard  parallels,  94;    time,   132. 

Stationer's  table,  147. 

Statement,  of  account,  46 ;  in  propor- 
tion, 243. 

Statutemile,67, 129. 

Stock  company,  221. 

Stocks  and  bonds,  221 ;  exchanges,  225. 

Study,  suggestions  for,  16. 

Subtraction,  10;  horizontal,  11;  of 
decimals,   25 ;    of  fractions,   56,   57. 

Surface,  98;  forms,  98;  of  cylinders, 
pyramids,  cones,  and  spheres,  107. 

Survey  of  land,  94. 

Surveyor's  measure,  68. 


Tariffs,  188. 

Taxes,  185. 

Term  discount,  173. 

Term  of  discount,  213;    table,  214. 

Term  policy,  237. 

Terms  of  ratio,  240. 

Time  certificates  of  deposit,  210. 

Time,  measures  of,  126 ;    difference  in, 

127 ;     compound    subtraction,    127  ; 

relative,  130 ;    and  longitude,  137. 
Tornado  insurance,  232. 
Township,  94. 
Trade  discount,  172. 
Tradesman's  table,  147. 
Trapezium,  100,  104. 
Trapezoid,  100,  102. 
Traveler's  checks,  146. 
Triangles,   99 ;   area  of,   102 ;    applicar 

tion  of  square  root  to,  80. 
Troy  weight,  137. 
True  discount,  215. 

Undivided  profits,  222. 
Unitate  addition  proof,  4. 
Usury,  194. 

Valued  insurance  policy,  232. 

Value,  unit  of,  141. 

Volume,  109 ;  of  rectangular  prisms, 
110;  non-rectangular,  123  ;  of  pyra- 
mids, 123 ;  cylinders,  cones,  and 
spheres,  124;  metric,  151. 

Weight,  measures  of,  137 ;   metric,  152. 
Wood  measure,  113;    metric,  152. 
Writing  decimals,  24. 


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The  Geography  of  Commerce 

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This  book  is  exceptionally  fortunate  as  well  as  unique  in  its  authorship.  Dr. 
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The  Geography  of  Commerce  gives  a  clear  presentation  of  existing  conditions 
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the  United  States  and  other  countries  are  given  special  prominence.  The  causal 
relations  of  physical  environment  to  men,  of  men  and  environments  to  products, 
and  of  products  to  trade,  are  treated  with  a  unity  that  makes  the  book  admirably 
suited  to  class  use. 

A  complete  working  equipment  and  a  list  of  books  for  further  consultation 
are  furnished.     Supplementary  questions  and  topics  are  also  supplied. 


PUBLISHED    BY 

THE   MACMILLAN  COMPANY 

64-66  Fifth  Avenue,  New  York 
BOSTON  CHICAGO  DALLAS  SAN  FRANCISCO 


Commercial  Correspondence  and 
Postal  Information 

By  Carl  Lewis  Altmaier 

Mr.  Altmaier's  work  supplies  two  present  needs,  a  text-book  for  school  use 
and  a  handbook  for  office  use.  In  the  first  place,  his  book  is  a  working  manual 
for  instruction  and  practice  in  letter  writing,  and  thus  it  furnishes  material  for 
practical  English  composition.  Correct  forms  of  letters  are  furnished,  after  which 
the  learner  is  asked  to  deal  with  situations  of  the  kind  actually  met  with  in  busi- 
ness correspondence.  The  treatment  of  correspondence  is  supplemented  by  a 
somewhat  detailed  account  of  postal  arrangements,  both  domestic  and  interna- 
tional. The  book  is  illustrated  with  photographs  of  documents,  reproductions  of 
actual  letters,  and  a  postal  map  of  the  world. 

Comprehensive  Bookkeeping 

By  Artemas  M.  Bogle 

A  few  of  the  points  that  commend  this  volume  are : 

I.  The  gradual  and  systematic  development  of  the  subject.  2.  Preliminary 
sets  for  drill  followed  immediately  by  more  concrete  sets  for  the  more  advanced 
work  of  the  student.  3.  Material  so  arranged  that  the  teacher  may  use  it  largely 
in  his  own  way.  4.  The  sets  so  arranged  that  short  exercises  or  longer  ones  may 
be  given  as  may  be  most  advantageous.  5.  Provision  for  drill  on  important 
points  and  at  the  place  where  needed,  thus  insuring  the  mastery  of  each  point. 
6.  Arrangement  such  that  at  almost  any  stage  previous  points  may  be  reviewed 
without  going  back  and  working  over  the  old  material.  7.  Clear,  concise  expla- 
nations. 8.  A  large  number  of  cross  references,  showing  the  connection  of  one 
portion  of  the  subject  with  another. 

Teacher's  Manual  to  Accompany  Comprehensive 
Bookkeeping 

By  Artemas  M.  Bogle 

This  book  contains  the  results  of  computations  required  by  the  regular  series 
of  exercises  given  in  Bogle's  "  Comprehensive  Bookkeeping."  These  tables,  giv- 
ing the  "  answers  "  which  should  be  right,  save  the  teacher  labor  in  checking  up 
pupils'  results.  The  forms  are  not  intended  for  models  but  only  as  results  to  save 
labor  by  the  teacher. 


PUBLISHED    BY 

THE  MACMILLAN  COMPANY 

64-66  Fifth  Avenue,  New  York 
BOSTON  CHICAGO  DALLAS  SAN  FRANCISCO 


UNIVERSITY  0T=^  <  ALIFO^ 


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RETURN  TO  DESK  FROM  WHICH  BORROWED 


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Renewed  books  are  subject  to  immediate  recall. 


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